Motion

Inertia

A body in motion tends to remain in motion; a body in rest tends to remain at rest. How to describe motion?

• Position: Where you are
• Velocity: 2 parts:
  • 1.) How fast you are going (speed)
  • 2.) Which way you are going (direction)
• Acceleration: How fast your velocity is changing: Speeding up, Slowing down, Changing direction

Newton’s law

• First Law: An object that is not subject to any outside forces moves at constant velocity, covering equal distances in equal times along a straight line path.
• Second Law: The force exerted on an object is equal to the product of the object’s mass times its acceleration. The acceleration is in the same direction of the force
  • F = ma
  • Easier to move a light object than a heavy one
• Third Law: For every force that one object exerts on a second object, there is an equal but oppositely directed force that the second objects exerts on the first object.

Gravity

A man standing on the ground, not moving, no acceleration. By Newton’s second law, 0 force. Does not mean no gravity. Think of all forces, there’s normal force from floor pointing upwards. These two forces cancel out.

Consider falling ball. \(F=mg=ma \implies g=a\). Acceleration independent of mass. Feather falls as fast as a lead brick.

Energy

is ability to do work/damage. Two energy:

• potential (stored): Higher has more potential energy.
• kinetic (moving): Faster moving has more kinetic energy.

Work

is how you transfer energy. work = force \(\times \) distance. If you are not pushing or it is not moving, then you are not working.

Momentum

is tendency to continue moving in a certain direction. More momentum: Tend to win in collisions

Momentum = Mass \(\times \) Velocity

Friction

How to push a block. Gravity = normal force (block not falling). Push force - friction = ma (the block accelerate)

Friction slows the motion. Push force needs to be greater than friction for motion to occur. Friction only resists the motion.

Two kinds of friction

1. Static friction: Not moving; Keeps the object in place (desk on the floor)
2. Kinetic friction: while moving

Generally static friction is bigger than kinetic hard to get the motion started

Friction force does work \(\to \) HEAT

Circular Motion

Constant speed, but the direction of velocity changes. It is always accelerating. Thus requires an inward force, pointing towards the center of the circle. \(F=mv^2/R \). Bigger radius, smaller force. Bigger velocity, larger force.

Car going around a corner/circle. No strings, where does that inner force come from? friction between tire and road. Well, there’s force pushing me towards the center, but I feel I am being thrown out of the car while turning left. Is there a force pushing me out of the car? No. Imagine the seats are slippery. Then the person is moving straight but the car is going left by Newton’s first law.

Demo here shows the conservation of angular momentum. If it keeps spinning, it doesn’t fall. If it slows down, then gravity takes over.

Another demo shows momentum of inertia. We cannot change the mass, but we change the distribution: by moving mass further away/closer to the center. Putting hands and legs, it becomes faster.

Bucket

Hold a bucket of water and rotate it around your head, the water doesn’t fall when the bucket is above your head. It goes fast enough, the force needed is greater than the gravity, then the water stays. If the gravitational force is smaller than that required for circular motion, the bucket will exert more force on the water to keep it moving in a circle.

Resonance

Demo: wine glass with straw inside. Round circle speaker at the back, amplifier. We can see straw going back and forth, and finally the glasses breaks.

Travelling Waves

Pulse on a string.

Can have a wave travelling down the string OR can have a standing wave.

Demo: fix a rope at one end and hold on the other end. The pulse hits that end and comes back upside down. If we pull tight, create more tension, it travels more faster. The pulse carries the energy.

• Transverse Oscillations: String moves up and down forming a wave that travels along the string, similar to waves in the ocean
• Longitudinal Oscillation:
  • Pressure (= Force divided by Area) is a force exerted over an area.
  • Strong force on a large area has same pressure as weak force on a small area
  • Sound is an example of this type of wave. Regions of low pressure and high pressure. Can see compression and expansion in different places.

Speed of a wave = Frequency \(\times \) Wavelength

Why does string move up and down? Restoring force from tension in the string

Doppler Effect

Change in Train Whistle.

• Train not moving, Wavelength \(L \), Frequency \(f \)
• Train moving right, towards you. Wavelength smaller, freq higher.
• Train moving left, away from you. Wavelength larger, freq lower.

Speed of sound constant. Waves get compressed or expanded

Pendulum

• String pulls the mass to the right
• The mass keeps moving, past the bottom and off to the right
• The string then pulls the mass to the left
• The mass goes back and forth
• Restoring force: always tries to bring the mass to equilibrium
• Simple Harmonic Oscillator
• Period of pendulum ( Time to go back and forth) only depends on length of the string
• Long string, long period, low frequency

Harmonic Oscillator

Mass on a spring

• Start with spring stretched
• Spring pulls mass to the left
• Spring gets squeezed and pushes the mass to the right
• Spring applies a restoring force to the mass (tries to place the mass so the string is unstretched and relaxed
• Frequency depends on mass and the stiffness of the spring
• Bigger mass, lower frequency. Stiffer spring, higher frequency

Resonance

Example: child on the swing and push the child. That’s pendulum. Because of the friction, eventually stop. If you take phys 112, then you will know the period changes if the amplitude becomes large enough.

You can’t push in any given time. Always push just as the swing moving away from you. Pushing does the work, which increase the mechanical energy. Here the kinetic energy gets increased, moving faster. At the top, lots of gravitational potential energy. If we push when it comes towards me, we decrease the kinetic energy.

Oscillators have a natural frequency at which they prefer to oscillate. This is the resonant frequency of the oscillator. If energy is applied to the oscillator at the same frequency, the amplitude of the oscillation will increase.

Standing Waves

string instruments. If string has two fixed ends, the waves will travel and hit the end and comes back flipped upside down. We got some waves travelling to the right, and some to the left. There are some interference. We see the sum of waves. String can’t move on two ends. For some specific freq, we see standing waves.

Demo: 5 hz is the fundamental freq, first harmonic. 10 hz we see the second. 15 hz is the third, 3 antinodes.

In guitar, we got fret. Changing the fret, we can adjust the length of the string. Speed constant, wavelength different so frequency different (harmonics)

Wine Glass

The sound makes the glass vibrating. Plastic cup will not likely break because energy could loss in the cup. Little energy loss in good wine glass for every oscillation of the cup.

• Rub finger along the rim of glass get friction
• Stick / slip causes rim of wine glass to move
• Size and shape of the glass determine the natural frequency
• Rubbing finger adds energy and makes sound louder
• Similar to bow on a violin

Bowl of wine glass moves in and out like a water balloon. This moves air back and forth, producing a pressure wave SOUND

Sounds from a Tube

In wind instruments, sound is little different. In the pipe, we have standing waves of the actual air.

• Pressure differences in tube provide a restoring force
• Simple harmonic oscillator
• Fundamental wavelength is twice length of tube
• Blow air across bottom of tube
• Adds energy to resonator, coupling complicated
• Air moves in and out at the ends

Demo: freq depends the length of the column of air. The temperature and density of gas affect the speed of sound.

Sound from a beer bottle

Behaves like a mass on a spring. Air in a bottle is the mass. Air in the large part of the bottle provides the restoring forces as it expands and contracts.

Adding liquid to the bottle decreases the size of the air compartment, changes restoring force and therefore the frequency

Loud Speaker

The pressure wave is created by the speaker cone moving back and forth.

Pressure

Gas

Inside the air, we have oxygen molecules, nitrogen. Can be single atoms or molecules. All molecules are moving fast, and bouncing off each other. Different from liquid. Solid, molecules are regularly aligned.

Pressure

Where does pressure comes from?

We have a bunch of gas particles bouncing around the box. Some of them are bouncing off the walls in the box. Molecules have mass. When they hit the wall, there’s a collision, some force applied to the wall. So there’s a force pushing outwards on the box. Rather than talking about the force pushing the box, we talk about the pressure, which is \(force/area \). That’s a property of a gas, which is independent of the size of the box/area.

How we can change the pressure? We can change the pressure by changing the density. Or temperature. Higher the temperature, faster those particles are moving, more kinetic energy, faster motion, more force, greater pressure.

Density is how many particles I have in certain amount of volume. More particles inside the box, more collisions with the walls, more force, higher pressure.

Ideal Gas Law

\[ PV = nRT \]

where P is pressure, V is volume, n is number of moles, R is constant (fudge factor), T is temperature. Pressure is proportional to density of particles times temperature.

Atmosphere

Why do we have atmosphere pressure. There’s gradual transmission between atmosphere and space. Gravity is giving us gravity air pressure.

Let’s take a little box of air. Each stacks on each other. The gas has some weight.

What does pressure really mean?

• Atmospheric pressure at sea level: 100,000 newtons per square meter, 100, 000 pascals (Pa) (~10,000 kg per square meter or 14.7 pounds per square inch)
• Pressure at Denver: 83,000 Pa (12.2 psi)
• Pressure at Chomolungma (Mount Everest): 33,000 Pa (4.9 psi)
• At sea level, equivalent to a bus on top of you or 1 ton per square foot
• Pressure difference between top and bottom of wing 2, 000 Pa (1/3 psi)

So we have ~10000kg air on us. Why we are not flattened into a pancake? It turns out air pressure presses us on all sides. Equally from all sides. We are made up of water. Water is compressible.

Hydraulic Pump

Buoyancy

Archimedes’ Principle: An object partially or wholly immersed in a fluid is acted upon by an upward buoyant force equal to the weight of the fluid it displaces.

Consider part of the fluid: Total Force = force up – force down – force of gravity = 0

If the object has a different mass, the force of gravity will be different

• if mass is greater than the fluid. Total Force is negative, object falls
• if mass is less than the fluid. Total Force is positive, object goes up
• if mass is the same Total Force = 0, object stays where is

Hot Air Balloon

Floating through the sky. Fire burner at the bottom, which heats the air. The bottom of the balloon is completely open. Air can come in and out. Why does hot air balloon go up? The density of the hot air balloon must be less than the density of the air. How’s that possible? The key is the fire. When we heat up the air inside the balloon, the temperature gets higher, the kinetic energy of the air particles gets larger. Then the pressure goes up, but the bottom is open. Air will then come out from the bottom. The density inside will decrease significantly.

pressure inside = pressure outside because of the hole on the bottom.

The density of the balloon then decreases. Pressure is proportional to density times temperature. Hot air has higher temperature and same pressure so density is lower. So weight of balloon is less than the air. Then balloon rises.

Regular Balloon

First, prof blows up a balloon. Pressure inside > pressure outside, because it needs to stretch balloon. We are doing work to inflate the balloon. Tension on balloon produces inward force, pressure of balloon provides outward force to balance.

Helium Balloon

Force due to gravity = weight of balloon + weight of helium

If weight of helium is less than weight of air (yes) and weight of helium + weight of balloon is less than weight of air (depends on balloon) then balloon goes up

Bubbles

A bubble is made up of soap and water. Surfactant molecules (SOAP)

Why do bubbles pop? Water inside stablizes the bilayer. As water vaporizes, it starts to thin the membrane. Until the membrane is not strong enough, then it corrupts.

Air Cleaners and Xerox

Electric Charge & Force

Two types: Electron is negative, Proton is positive, (no smaller electric charges).

Benjamin Franklin (1700s): Silk rubbed glass is positive, Fur rubbed rubber is negative,

Charge is measured in Coulombs (C). 1 Coulomb = 6.28x1018 charges

• Neutral objects have both negative and positive charges (an equal number of both)
• Something positively charged has more positive charges than negative
• Something negatively charged has more negative than positive

\[ F= {kq_1 q_2\over R^2} \]

Voltage

• Power supply (battery) pushes current around the circuit
• Battery keeps one side positive and other negative
• Work is done in lighting the bulb
• This work comes from the charges in the circuit which in turn comes from the battery
• Voltage measures the potential energy of a unit of charge at a particular location

Higher voltage, positive charge more potential energy, releases more energy into the light bulb, light bulb glows better.

Dust

Dust is small and light, moving around in the air. Can be all sorts of different things: rock, dirt, organic matter in tiny pieces.

Soot: carbon, organic material that has been burned imperfectly, oily, greasy, tarlike

Ash: powdery non‐combustible residue of a fire

Basically neutral particles floating in the air

Then we have dust filters: Can filter dust by passing air through small pores: air goes through, large dust particles get trapped.

Two ways to stop the particles.

• Inertial impaction. Imaging filter being lots of fibers intersecting each other, with pores in between. Larger pieces of dust have velocity, traveling in straight lines. At some point, it hit the filter and stick to it.
• Diffusive motion/Brownian motion. Small particles zigging-zagging.

Not good with intermediate size particles. Small dust goes through clogging slows air flow, then air doesn’t move out.

Electronic Air Cleaner

Recall dust particles are neutrally charged. We could somehow charge the dust. For example, we give the dust negative charge. Then we pass the dust through positively charged cylinders. Then it sticks to the container. Then bang the wall to get the dust off.

How do we charge dust? Take a power supply, which can generate large voltage. Connect one end to ground, the other end to a pin (wire). Wire starts to emit charges, which spill out to the air. The charges repel each other and eventually end up on the tip where they are very close together. When the charge has built up to a large enough value, the electrons jump off the pin on their own.

Then corona discharge: A corona discharge is an electrical discharge caused by the ionization of a fluid such as air surrounding a conductor carrying a high voltage (from wiki). Potential energy is released when the electrons fly off the in: Light is emitted

Lightning Rod

is a stick sitting at the top of the building/tower. Attach that rod to the ground with a wire. Ground is a big source of charge: can release/store a lot of charge, and get charged. In a lighting storm, we get clouds, which are charged. Eventually, those charges want to go to different places: other clouds. We don’t want the lighting hit us/the house. So we make our ground less attractive to hit by lightning. Try to discharge any charge on the ground around us, which is where we use lightning rods. Any charge on the ground will go through wire and get corona discharge coming off the lightning rod, which will discharge the charge around the building. Then because it is neutral, less attractive for the big spark from the cloud to hit the rod.

Electrostatic Precipitator

• neutral dust
• Power supply between walls and corona wires
• Corona discharge from the negatively charged wires releases electrons into the air that then charge the dust particles, making them negatively charged
• This negatively charged dust is then attracted to the positive walls, trapping the dust
• The electrostatic force is much stronger than gravity and any air currents

Other Dust Cleaners

Ion generator: Charges dust but does not trap it. Instead it sticks to the neutral walls of the room or the furniture. Make the room more dirty. Charged dust particle polarizes the neutral surface so that they attract

N95 Masks

Two kinds of standard face masks. One is like filter. Air goes through the fiber of the filter, common cloth. What goes through quite easily are intermediate sized of moisture.

N95: 95% gets blocked by it. Out layers block large pieces. Thick layer of filter: bunch of fibers. Then the key to N95 is the additional layer behind here, made up of particular type of polymer. Looks similar to the first one, but it is stiffer than the first thick layer. There are charges in these fibers. Then those intermediate sized particles can be electrostatically attached to the filter.

Xerox Machines

Ink is made up of toner particles (plastic particles that are coloured). When they melt, they stick to the paper. It works with electrostatics.

Photoconductor: in dark, insulator. In light, conductor. Lots of light, good conductor. Less light, less of a conductor (more resistance to charge flow)

• Place charge on both sides of the photoconductor
• illuminate one spot, that spot becomes a conductor, charge can move and gets neutralized
• There is no isolated charge where the light shines

Laser Printer

By KDS4444 - Own work, CC BY-SA 4.0, Link

• Use laser to scan an image onto a photoconductive drum
• This creates a charge image on the drum that can be covered with toner
• This toner is then transferred to the paper, creating the image

Electricity

Electric Circuits

How do charges move? Electrons orbit the nucleus Some electrons break free of the nucleus and can move freely through the metal. Conduction electrons.

In a conductor, the atoms inside release electrons, then electron is free to walk around the entire structure. Those free electrons can be moved around. We can use electric field to move electrons. Any time we have a current flow, we have the negative charges, electrons moving around the conductor.

In an insulator, charges don’t move. There are no free conductor electrons. All electrons are bound to the atoms or nucleus.

In an electric circuit, current moves from positive to negative. Actually electrons travel from - to +

Resistance and power

Resistance is a measure of the opposition to current flow in an electrical circuit.

Ohms Law: \(V=IR \). Voltage = current \(\times \) resistance

Constant voltage:

• small resistance gives big current
• big resistance gives small current

Power is energy released per time Big power, lots of energy released per second Small power, little energy released per second

\(P=VI \). Power = voltage \(\times \) current.

Thus \(P=I^2R \)

Magnet & Current & Electricity

Direct/Alternating Current: Direct current is what we have seen so far, current travels in one direction. For simple model, voltage difference is constant. Alternating current, direction of the current keeps changing. For example, \(V=k\cdot \sin(t) \). Different frequencies in different places.

Magnets: A little bar, one side north, one side south. North and north repel, south and south repel, north and south attract. You can make magnets out of magnetic material. You can also make magnetic field with a wire, like electromagnet, current in a coil. Magnet field around the earth.

Magnet and Electricity. Electric current can make magnetic field. Wind the wire around the soft magnetic material, then it can take things up. Hard magnetic materials: once magnetize it, it will stay magnetized. Soft magnetic materials: it doesn’t behave like a magnetic, paperclip is an example. Iron bar has magnetic domains that are normally randomly oriented when there is not external field. Domains oriented with the magnetic field from the coil grow (the others shrink) This amplifies the total magnetic strength of the electromagnet. When we remove the external magnetic field, soft magnetic material becomes randomized again.

Change magnetic field induces an electric field. Move permanent magnet in and out of coil to change magnetic field inside coil. Changing magnetic filed inside coil induces voltage across the coil. This voltage (and current) can power an electrical load (light bulb)

Lenz’s Law

(楞次定律) Current induced by a changing magnetic field always produces a magnetic field that opposes the change.

push magnet into coil \(\implies \) coil tries to push back. I need to do work, which is transferred into electrical energy. This can be used to make a generator.

Lenz’s Law: harder to turn when there is a large load

Another application: transformer: convert between different voltage.

The ratio of the two voltages is the same as the ratio of the number of turns on the coils.

\[ V _ {in} / V _ {out} = \frac{\text{number of turns in primary}}{\text{number of turns in secondary}} \]

House Power (skipped)

Fuses and circuit breaker

Fuse wire heats (I2R) eventually melts and breaks the circuit \(\to \) protects the light bulb

Circuit Breaker, reusable

• Bimetallic strip: two strips of metal that thermally expands at different rates
• Bimetallic strip heats (I2R) bends and pulls the contacts apart protects the light bulb

They are designed to protect the house. The wires do not overheat, the houses do not burn down.

Ground Fault Circuit Interrupter (GFCI)

is designed to protect you from being hurt. wiki

If the currents are not perfectly no magnetic field is induced in the coil balanced (5mA difference), the coil will sense the difference and activate the solenoid, opening the switch and turning off the power, There are also mechanical pieces that make certain power can not turn on again unless the reset button has been pushed.

Live and neutral wires pass through a coil. If there is no leakage current (no short), the two currents cancel out and no magnetic field is induced in the coil.

Radio & TV

E & M Review

Like charges repel; unlike charges attract. Electric field lines run from positive to negative charges, and a charged particle experiences a force along those field lines. The new piece of physics for this week is that a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field. These relationships, combined with everything covered in the electricity week, form the complete set of Maxwell’s equations — the foundation of all radio and television technology.

Antenna

An antenna is simply a piece of conducting wire or a metal pole. When alternating current flows through it, electrons oscillate back and forth. Because those electrons are accelerating, they produce a changing electric field, which produces a changing magnetic field, which produces another changing electric field — and the disturbance propagates outward through space as an electromagnetic wave, carrying energy away from the antenna at the speed of light.

A receiving antenna works in reverse. The oscillating electric field of the incoming wave pushes the electrons in the antenna wire up and down, creating a tiny alternating current that the radio’s circuitry can detect and amplify. The receiving antenna must be oriented the same way as the transmitting antenna: a vertical transmitting antenna produces a wave with a vertical electric field, and only a vertical receiving antenna will have charges that respond to that field. Turn the receiving antenna horizontal and you pick up nothing.

Electromagnetic Waves

In an electromagnetic wave, the electric field \(\mathbf{E}\) and the magnetic field \(\mathbf{B}\) oscillate perpendicular to each other and to the direction of travel. All electromagnetic waves travel at the speed of light, \(c = 3 \times 10^8 \text{ m/s}\) in vacuum. Frequency and wavelength are related by:

\[c = f \lambda\]

The electromagnetic spectrum spans an enormous range: radio waves at the low end (long wavelengths, low frequencies), then microwaves, infrared, visible light — the narrow slice our eyes respond to — ultraviolet, X-rays, and gamma rays at the high end. All are the same phenomenon; only the frequency differs.

Tank Circuit

A tank circuit (also called an LC circuit) consists of a capacitor and an inductor (a coil of wire) connected in a loop. Energy sloshes back and forth between them in a way that is exactly analogous to a pendulum. The capacitor stores energy in its electric field and discharges through the inductor; the inductor stores energy in its magnetic field and then recharges the capacitor; and so on. The circuit oscillates at a natural resonant frequency:

\[f = \frac{1}{2\pi\sqrt{LC}}\]

where \(L\) is the inductance and \(C\) is the capacitance. Changing either component tunes the resonant frequency — exactly what the tuning dial on a radio does. A variable capacitor whose capacitance you adjust by turning a knob shifts the resonant frequency to match whichever station you want to receive.

The tank circuit also amplifies weak signals through resonance. If you drive the circuit at exactly its resonant frequency — like pushing a child on a swing at just the right moment each cycle — the amplitude of oscillation builds up. Drive it at the wrong frequency and it barely responds. This selectivity is what lets a radio pick out one station from dozens broadcasting simultaneously.

Radio Transmitter and Receiver

A simple transmitter connects a tank circuit to a transmitting antenna. The oscillating current in the tank circuit drives electrons up and down the antenna, radiating an electromagnetic wave at the tank circuit’s resonant frequency. The transmitter must continuously supply energy to compensate for resistive losses in the circuit and, more importantly, for the energy that carries away in the electromagnetic wave itself.

At the receiver end, the antenna is connected to another tank circuit — tuned to the same resonant frequency. The tiny oscillating current the incoming wave induces in the antenna adds energy to the tank circuit at just the right moment each cycle, and the amplitude of oscillation builds until it is large enough to be easily detected. Signals at other frequencies add energy at the wrong phase and produce no net build-up.

For AM radio the carrier wavelength is roughly 300 m, making a quarter-wave antenna about 75 m tall. Those tall broadcast towers you see in rural areas literally are the antenna — the entire steel structure is the radiating element. A portable receiver cannot carry a 75 m rod, so instead it uses a coil of wire. Because the magnetic field of the wave is perpendicular to its electric field, orienting a coil to capture the changing magnetic flux works just as well. Rotating an AM radio affects signal strength for exactly this reason: the coil picks up more or less of the oscillating magnetic field depending on its orientation relative to the transmitter.

Driving under large power lines produces a characteristic buzz on an AM radio. Those lines carry alternating current at 60 Hz — and accelerating charges at 60 Hz radiate a 60 Hz electromagnetic field. Near the towers the field is strong enough to overwhelm the tank circuit’s frequency selectivity, forcing the receiver to follow 60 Hz rather than the station’s carrier frequency. Move far enough away and the power-line field weakens, the tank circuit reasserts its resonance, and the station comes back in.

AM and FM

There are two main methods of encoding audio onto a radio carrier wave. In amplitude modulation (AM), the amplitude of the high-frequency carrier is varied in proportion to the instantaneous value of the audio signal. In frequency modulation (FM), the carrier amplitude stays constant but its frequency is varied slightly above and below the centre frequency.

AM is technically simpler but susceptible to noise, because most natural and electrical interference manifests as amplitude variations — exactly what AM uses to carry the signal, so the receiver cannot distinguish interference from programme content. FM is far more noise-resistant: any spurious amplitude variations can be stripped off by a limiter before the signal is decoded, since all the audio information is in the frequency, not the amplitude. This is why FM sounds noticeably cleaner than AM.

The trade-off is bandwidth. An AM signal occupies only a few kHz; an FM signal needs about 200 kHz. Fewer FM stations therefore fit in the allocated spectrum, and they are spaced farther apart on the dial. The FM broadcast band runs from 88 to 108 MHz; the AM band occupies 530–1700 kHz.

A crystal radio needs no battery or power supply at all. It uses a coil antenna connected to a tank circuit, with a semiconductor crystal acting as a diode detector. The tiny energy captured from the radio wave is just enough — after resonant amplification in the tank circuit and detection by the diode — to drive a high-impedance earphone. It was the first form of home radio receiver and works entirely on the energy delivered by the incoming electromagnetic wave.

Theremin Demos

The theremin is one of the earliest electronic musical instruments, and it is played without being touched — the performer simply moves their hands through the air near the instrument. A theremin contains two oscillator circuits, each built around a tank circuit. One controls pitch; the other controls volume. The player’s hand acts as one plate of a capacitor: the instrument’s antenna rod is the other plate, with air as the dielectric. Moving the hand closer to or farther from the antenna changes the capacitance, which shifts the resonant frequency of the tank circuit, which changes the pitch or volume of the output.

The homemade version uses two TV-antenna rods as the capacitor plates, with simple electronics converting the capacitance change into an audio frequency. The Moog theremin is a commercially designed instrument with much finer control. Even a single vertical rod can act as a capacitor if you think of the second plate as sitting “at infinity” — the hand still changes the effective capacitance and thus the frequency.

Cathode Ray Tube

The cathode ray tube (CRT) is the display technology inside old-fashioned deep televisions and computer monitors. The front face is a thick piece of glass coated on the inside with a phosphor — a chemical that emits visible light when struck by electrons. At the back of the tube is an electron gun that produces, focuses, and steers a beam of electrons.

The electron gun has four main parts. The cathode is a hot wire or pin: heat gives electrons enough energy to boil off the surface. You can sometimes see a faint orange glow through the ventilation slots at the back of an old TV — that is the glowing cathode. Next comes the grid, a plate with a small hole and a variable negative voltage. The more negative the grid, the more it repels electrons and the fewer pass through; varying the grid voltage varies the intensity — and thus the brightness — of the beam. Beyond the grid, the anode is a positively charged disk with a hole; electrons are attracted to it and accelerate, gaining kinetic energy. Finally, focusing coils around the neck of the tube squeeze the beam into a sharp point on the phosphor.

Steering is accomplished by electromagnets. Two pairs of coils around the tube neck produce magnetic fields that deflect the electron beam left/right and up/down. A moving electron is a moving charge, and a moving charge in a magnetic field experiences a force perpendicular to both its direction of travel and the field direction — unlike electric or gravitational forces, which act along the field. Increasing the current through the coils increases the magnetic field and deflects the beam further.

To produce an image, the beam rasters across the screen: it sweeps slowly from left to right, snaps back instantly to the left, drops down one line, and sweeps again, covering the entire screen many times per second. As it sweeps, the grid voltage is modulated to create bright and dark spots. Analog television used interlaced scanning: the odd-numbered lines were drawn on one pass, the even-numbered lines on the next. Each half-frame was refreshed at 60 Hz, giving the visual impression of 30 complete frames per second — fast enough that the eye perceives smooth, continuous motion.

Colour Television

A black-and-white CRT has a single phosphor that glows white. Colour television works by coating the inside of the screen with millions of tiny dots — or, in Sony Trinitron tubes, fine vertical stripes — in three different phosphors that glow red, green, and blue when struck by electrons. Mixing these three colours at different intensities can produce any colour in the visible spectrum. The electrons themselves carry no colour; it is entirely the choice of phosphor that determines what colour of light is produced.

The tube contains three separate electron guns — one for each colour. To prevent a beam intended for, say, the red dots from accidentally striking a green or blue one, a shadow mask sits just inside the face of the screen. It is a thin sheet of metal pierced with thousands of tiny holes. The geometry is arranged so that, looking through any given hole, each gun can see only its own colour of phosphor dot. The three beams are steered together by the same set of deflection coils, always pointing at the same region of the screen, but through the shadow mask each ends up illuminating only the correct dots.

The shadow mask is made of a soft magnetic material, typically iron. Bringing a permanent magnet close to the screen magnetizes it, and the resulting stray field bends the electron beams off-course so each no longer hits its intended colour phosphors — the image becomes splotchy with shifted colours. Fixing this requires degaussing: applying a strong alternating magnetic field that is then slowly tapered to zero, leaving the shadow mask fully demagnetized. Older TVs had to be taken to a repair shop for this; many high-quality CRT monitors had a degauss button on the front panel. Pressing it produced a characteristic loud thump and brief swirling of the image as the degaussing coil fired. Black-and-white televisions had no shadow mask and were therefore immune to magnet damage — a distinction that is irrelevant now that CRT displays have largely disappeared.

Microwaves & CD Players

Waves and Light

Waves have two fundamental properties: wavelength — the repeat distance from one crest to the next — and frequency — how many complete cycles pass a given point per second. These are linked to wave speed by the relationship \(v = f\lambda\). For a wave on a string, speed depends on the string’s tension and mass density. For electromagnetic waves, the situation is different: they require no medium at all and travel through vacuum at a universal constant, \(c \approx 3 \times 10^8\) m/s. In a material such as glass, the speed is lower — roughly \(c/1.5\) for typical glass — but the frequency cannot change as the wave crosses from one medium to another. Since \(v = f\lambda\) and \(f\) is fixed, the wavelength must decrease when the wave slows down.

Visible light is simply the narrow band of electromagnetic radiation our eyes happen to be sensitive to: wavelengths from about 0.4 microns (violet) to about 0.7 microns (red). Different colours correspond to different frequencies; in a vacuum they all travel at the same speed. When white light — which is a mixture of all visible wavelengths — passes through a glass prism, each colour travels at a slightly different speed in the glass, so each bends by a slightly different angle at the surface. The prism therefore fans the colours out into the familiar rainbow spectrum.

The full electromagnetic spectrum spans many orders of magnitude on either side of visible light. Going to longer wavelengths (lower frequencies) from visible: infrared, then microwaves, then radio waves — both radio and visible light pass through the atmosphere relatively freely, which is why they are useful for broadcasting and why our eyes evolved to use visible light. Going to shorter wavelengths: ultraviolet, X-rays, gamma rays — these are largely blocked by the atmosphere.

Microwaves

Microwaves occupy the wavelength range from roughly one millimetre to one metre, corresponding to frequencies from about 300 MHz to 300 GHz. They are electromagnetic waves in every sense — oscillating electric and magnetic fields propagating at the speed of light — differing from visible light or radio only in their wavelength. The name simply reflects that they are shorter than radio waves; the physics is identical.

The frequency used in a household microwave oven is 2.45 GHz, corresponding to a wavelength of about 12 cm. This frequency was not chosen because of any special resonance with water molecules — rather, it was a frequency that was already allocated for industrial, scientific, and medical use and happened to interact well with polar materials. (It is also close to the 2.4 GHz band used by many Wi-Fi routers, which is why turning on a microwave can sometimes interfere with a wireless network.)

The key to microwave cooking is the interaction between the oscillating electric field and polar molecules — molecules in which the electric charge is not distributed symmetrically. Water is the prime example: the oxygen atom draws electron density away from the two hydrogen atoms, leaving the hydrogens slightly positive and the oxygen slightly negative. In an oscillating electric field, the water molecule experiences a torque trying to align it with the field; as the field reverses 2.45 billion times per second, the molecule tries to keep flipping back and forth. This continuous agitation is absorbed as thermal energy — the temperature rises.

This mechanism has important consequences. Frozen water — ice — does not heat efficiently in a microwave because the molecules are locked in a rigid crystal lattice and cannot rotate freely; the electric field cannot do work on them. This is why a frozen burrito can emerge from the microwave with some regions still ice-cold while others are scalding hot: a small pocket of liquid water or oil absorbs energy rapidly and heats up, then slowly conducts heat outward into the still-frozen regions. Pure non-polar hydrocarbons, like a simple alkane, similarly fail to heat — it is the polar molecules (including the fatty acid head groups in vegetable oil) that couple to the field.

The Magnetron

Radio waves at megahertz frequencies can be generated with an ordinary LC tank circuit: a coil and a capacitor oscillating back and forth. For 2.45 GHz microwaves, the required components are far smaller — a miniaturized tank circuit whose inductor is just a half-loop of metal and whose capacitor is formed by the two ends of that loop coming close together. A magnetron is an array of eight such tank circuits arranged in a cylinder with cylindrical symmetry, machined from a single block of copper. All eight circuits share conductors with their neighbours and oscillate at the same frequency in phase.

To sustain the oscillation, a central hot filament wire boils off electrons. Without any magnetic field, those electrons would simply accelerate straight outward toward whichever capacitor plate happens to be positively charged at that instant, and the effect would just cancel the existing charges rather than reinforce them. The ingenious solution is to add a strong magnetic field aligned along the axis of the cylinder. The magnetic force on the moving electrons is perpendicular to their velocity, so instead of flying straight outward they curve into arcing trajectories. With the field tuned correctly, the electrons curve and land precisely on the negatively charged capacitor plate each half-cycle — continuously adding charge and pumping energy into the oscillation rather than quenching it. A small loop antenna extracts the microwave signal from the changing magnetic field inside one of the inductors and feeds it into the oven cavity.

Microwaves and Metal

The inside of a microwave oven is lined entirely with metal — the walls, ceiling, and floor are all conducting. Rather than absorbing microwaves, the metal reflects them: the oscillating electric field drives a surface current that re-radiates the wave back into the cavity. This reflection confines the energy inside the oven.

Multiple reflections create standing waves in the cavity, just as reflections in an organ pipe create standing sound waves. At the antinodes the electric field is strong; at the nodes it is weak. Food placed at a node heats poorly; food at an antinode heats rapidly. This is why most microwave ovens include a rotating turntable — to average out the standing-wave pattern and produce more uniform heating.

Metal objects placed inside the oven must be treated with care. Thick metal has low resistance: induced currents flow easily, little heat is generated, and the object survives fine. Thin metal — the gold rim on a ceramic mug, a twist tie, crinkled aluminium foil, or the metallized coating on a shiny balloon — has much higher resistance, so the same induced current generates substantial heat. Corners and sharp tips concentrate the electric field further, which can produce visible sparks. If flammable material is nearby, this can start a fire. The thin metal film on a CD, subjected to microwave heating, vaporizes and scorches the plastic beneath it.

The door of a microwave oven appears transparent because it contains a metal mesh with holes much smaller than the 12 cm microwave wavelength: the microwaves cannot pass through openings smaller than their wavelength, so they are effectively blocked, yet the holes are large compared to the wavelength of visible light, so you can see inside. Two metal grids with their slots oriented at right angles form a complete electromagnetic barrier in the same way that two crossed polarizers block light.

The Microwave Oven

Inside a microwave oven the key components are the magnetron, a high-voltage transformer, and a set of safety interlocks. The transformer steps the household 120 V up to the few thousand volts the magnetron filament requires. The magnetron’s antenna protrudes through a hole in the metal cavity wall to inject microwaves.

Three separate switches monitor the door. One switch confirms the door is closed and permits operation; a second confirms it is open and prevents operation; a third is a kill switch that physically short-circuits the transformer if the door is open while power is somehow applied. If the kill switch ever fires, it blows a fuse and renders the oven permanently inoperable — a deliberate self-destruct to ensure that microwaves can never reach a person standing in front of the open door. A thermal cutoff switch on the cavity wall also shuts down the magnetron if the oven overheats.

Geometric Optics

A converging lens — thicker in the middle than at the edges, as in a magnifying glass — takes parallel incoming light and bends all of it to a single focal point. The smaller the spot the better, but there is a fundamental limit: the smallest focus achievable is approximately equal to the wavelength of the light. This limit will turn out to be crucial for determining how much data can be stored on a CD.

Mirrors obey the law of reflection: the angle of incidence equals the angle of reflection, both measured from the normal to the surface. A flat mirror produces an upright, life-size image. To see your full body in a flat mirror, the mirror need only be half your height — and this remains true regardless of how far you stand from the mirror. A parabolic (concave) mirror behaves like a converging lens, focusing parallel light to a point. Flashlights use parabolic mirrors in reverse: a bulb placed at the focal point sends light backward, which the mirror redirects into a parallel beam. A convex (diverging) mirror spreads light outward and produces a wide-angle, reduced image — familiar from security mirrors and the backs of spoons.

When light travels from a dense medium (glass) into a less dense medium (air) at a steep angle, it can undergo total internal reflection: instead of passing through the interface, it reflects perfectly back into the denser medium. This is the principle behind fibre optics — light injected into one end of a thin glass fibre bounces repeatedly off the inner surface and travels enormous distances with very little loss, never escaping through the side walls.

The human eye forms images by refracting light at the curved front surface and through the crystalline lens onto the retina. Nearsighted (myopic) eyes focus parallel light from distant objects to a point in front of the retina, producing a blurred image; a diverging (concave) spectacle lens pre-diverges the incoming light so the eye can then bring it to focus on the retina. Farsighted (hyperopic) eyes focus the image behind the retina; a converging (convex) lens corrects this by adding extra convergence.

Polarizers and Liquid Crystal Displays

Light is a transverse electromagnetic wave: the electric field oscillates perpendicular to the direction of travel. In ordinary (unpolarized) light the field oscillates in all orientations equally. A polarizer passes only the component of the electric field aligned with it and absorbs the rest. A single polarizer therefore transmits about half the intensity of unpolarized light. When two polarizers are stacked with their transmission axes at 90° to each other — crossed polarizers — no light passes through at all: the first selects one linear polarization, and the second is oriented exactly to block it.

Liquid crystals are materials that flow like a liquid but retain some of the orientational order of a crystal. In the nematic phase, the elongated rod-like molecules are positioned randomly (like a liquid) but all point in roughly the same direction (like a crystal). The orientation of the molecules can be controlled by surface treatments (fine scratches on the glass plates that confine the material) or by an applied electric field.

A twisted-nematic LCD pixel sandwiches a thin layer of nematic liquid crystal between two glass plates whose surface scratches are oriented 90° to each other. The molecules near one plate lie along one direction; those near the other plate lie along the perpendicular direction; in between, the molecular orientation gradually twists through 90°. This twisted arrangement rotates the polarization of light passing through it by exactly 90°. The sandwich is placed between two crossed polarizers. Without voltage: light passes through the first polarizer, its polarization is rotated 90° by the liquid crystal, and it can now pass through the second (crossed) polarizer — the pixel appears bright (or transparent). With voltage: the electric field straightens the molecules and destroys the twist; the polarization is no longer rotated, and the second polarizer now blocks the light — the pixel goes dark. A tiny colour filter — red, green, or blue — over each pixel produces colour images.

How to Make a CD

A compact disc encodes audio as a sequence of ones and zeros. The first step is digitizing the audio: the height of the sound wave is sampled at regular time intervals and rounded to the nearest value on a discrete scale. Audio CDs sample 44,100 times per second (adequate to faithfully represent frequencies up to about 20 kHz, the limit of human hearing) and quantize each sample to 16 bits — 65,536 possible levels — giving each sample a number that is then expressed in binary.

Binary is base-2 arithmetic: instead of the ten digits 0–9, there are only two — 0 and 1. Any integer can be expressed as a string of bits. A 16-bit sample can therefore represent 2¹⁶ = 65,536 distinct amplitude levels. The entire audio stream is thus reduced to a long sequence of 1s and 0s, which can be physically represented on the disc as two detectable states.

Those two states on a CD are pits and lands (flat regions). Manufacturing begins with a glass master coated in photoresist. A laser traces the spiral data track, exposing the resist wherever a pit should appear. Chemical development removes the exposed resist, leaving a pattern of raised and recessed regions. Metal is evaporated onto this master to produce the mother — a metal negative of the data surface. From the mother, further metal depositions produce the father and eventually the stamper, a robust metal plate whose surface is the mirror image of the finished disc. Molten polycarbonate plastic is then injected between two stampers, cooled, and ejected: a clear disc with microscopic pits moulded into one surface. A thin reflective metal coating is evaporated on, a protective lacquer layer is applied, and the label is printed on top. The data layer is only micrometres below the label surface — far more fragile than the clear plastic side.

The spiral track on a CD is about 5.4 kilometres long. The pit depth is approximately one quarter of the laser wavelength, and pit width is about half a micron — set by the diffraction limit of the infrared laser used to read it.

CD Player Optics

Reading a CD requires detecting whether the laser is reflecting off a flat land or a pit. The key is interference. Light reflecting off a pit travels a slightly different path length than light reflecting off the surrounding land. If the pit depth is exactly one quarter of the laser wavelength, then the round-trip path difference between the two reflected beams is exactly half a wavelength — the condition for complete destructive interference. A flat land reflects strongly (constructive interference); a pit reflects weakly (destructive interference). Ones and zeros are extracted from this alternating pattern of bright and dark reflections.

The optical path inside a CD player sends the laser beam through a beam splitter, a quarter-wave plate (which rotates the polarization on the outward and return trips so that reflected light is directed to the detector rather than back into the laser), and a focusing objective lens. The lens must be continuously adjusted — moved vertically to keep the focus precisely on the metal layer, and moved laterally to track the spiral. A voice-coil actuator (a small coil in a magnetic field, like a loudspeaker motor) provides fast, fine corrections to both focus and tracking.

The storage capacity of an optical disc is limited by the minimum achievable spot size, which is approximately equal to the laser wavelength. A smaller wavelength allows tighter track spacing and smaller pits, fitting more data into the same area. This is the entire story of the progression from CD to DVD to Blu-ray: each generation uses a shorter-wavelength laser.

Format	Laser colour	Wavelength	Capacity
CD	Infrared	~780 nm	700 MB
DVD	Red	~650 nm	4.7 GB
Blu-ray	Blue-violet	~405 nm	25 GB per layer

The technology is identical across all three formats; only the laser colour changes, and everything else — the track pitch, the pit size, the optical path — scales with the wavelength accordingly.

Alternate Energy

Generators and Motors

A generator converts mechanical energy into electrical energy. At its core is a coil of wire rotating inside a magnetic field — or equivalently, a magnet spinning inside a stationary coil. As the magnet rotates, the magnetic flux through the coil changes continuously, and by Faraday’s law this induces a voltage across the coil. Connect a load — a light bulb, a motor — and current flows. The mechanical work required to turn the generator against the electromagnetic braking force is what supplies the electrical energy. You can feel this directly with a hand-crank generator: it turns easily with no load, but the moment you close the circuit and light a bulb, it takes real effort to keep cranking.

A wind turbine is simply a generator driven by wind. The spinning blades capture kinetic energy from moving air and deliver it to a shaft; the shaft turns the generator, and the generator produces electricity.

An electric motor is the same device running in reverse: supply alternating current to the coil, and the changing magnetic forces cause the central magnet to spin. The geometry of the motor is identical to the generator; which direction the energy flows depends only on whether you are supplying mechanical energy to produce electricity, or supplying electrical energy to produce rotation.

For a DC motor to work — powered by a battery rather than alternating current — the direction of the current through the rotating coil must reverse every half-turn to keep the torque pushing the same way. This is accomplished by a commutator: a split-ring contact that swaps the coil connections every 180°, reversing the current at exactly the right moment. Brushes — spring-loaded contacts that slide against the commutator — carry current into the rotating coil. You can see tiny sparks at the brushes each time the commutator switches; this is a fundamental feature of DC motors.

The Gasoline Engine

A four-stroke engine converts the chemical energy of burning fuel into rotational mechanical energy through four sequential piston strokes.

In the intake stroke, the intake valve opens and the piston travels down, drawing a mixture of air and vaporized fuel into the cylinder. In the compression stroke, both valves close and the piston travels up, compressing the mixture to a fraction of its original volume. At the top of the stroke, the spark plug fires — ignited by a high-voltage pulse from a small generator on the flywheel — burning the fuel-air mixture and causing rapid expansion. In the power stroke, the expanding gases push the piston forcefully down; this is the only stroke that delivers energy to the crankshaft. Finally, the exhaust stroke opens the exhaust valve and the piston pushes the burned gases out.

The crankshaft converts the piston’s up-and-down motion into rotation, just as a bicycle crank converts the rider’s pushing into wheel rotation. A gear connects the crankshaft to the camshaft, which turns at half the crankshaft speed and controls the precise opening and closing timing of the valves through cams and pushrods.

The engine displacement — the number in litres you see in a car’s specifications — is the total swept volume of all cylinders together. A four-cylinder 2-litre engine has four cylinders of 0.5 litres each. More cylinders mean smoother power delivery because the power strokes are staggered around the crankshaft rotation rather than all occurring at once.

Hybrid Automobiles

In a conventional car, energy is lost in three main ways: air drag, friction in the drivetrain and engine, and — most importantly for stop-and-go city driving — braking. Every time you brake, all the kinetic energy the engine spent fuel to build up is dissipated as heat in the brake pads. This energy is completely wasted.

A hybrid vehicle recovers much of that braking energy through regenerative braking: instead of clamping brake pads onto a disc, the car connects the wheels to an electric motor-generator. The spinning wheels drive the generator, which produces electricity and simultaneously slows the car. That electricity charges the onboard battery. When the car next accelerates, the battery powers the electric motor to help drive the wheels — energy that was about to be wasted as heat in the brakes is instead stored and reused.

A typical hybrid system contains: a gasoline engine, an electric motor-generator (which serves as both motor and generator depending on the situation), a battery pack, and a power-splitting device that manages energy flow among all three. During gentle city driving, the car can operate on electricity alone from the battery. During hard acceleration, both the engine and the motor drive the wheels simultaneously — this means the gasoline engine can be smaller than it would otherwise need to be. On the highway, the gasoline engine runs alone while the battery sits at full charge, ready for the next stop.

The big efficiency gain is in city driving precisely because there are many braking events: more energy recaptured means less fuel burned. On the highway, where there is little braking, the hybrid advantage is smaller — though the smaller engine size does still help.

Formula One racing cars illustrate the same principle at the extreme end: they are officially called power units now because they combine a turbocharged internal combustion engine with an energy recovery system that captures both braking energy and exhaust heat energy, storing it electrically and deploying it for acceleration.

Electric Cars and Batteries

A battery stores chemical energy and releases it as electrical energy on demand. In an alkaline battery (the standard AA or AAA cell), two chemicals are separated by a membrane: zinc on one side, manganese dioxide on the other. The membrane allows water molecules and hydroxide ions (OH⁻) to pass through but blocks electrons. When the battery terminals are connected by a wire, a chemical reaction occurs: zinc reacts with hydroxide ions to produce zinc oxide, water, and — crucially — free electrons. Simultaneously, manganese dioxide reacts with water and electrons from the external circuit. The electrons thus flow through the external circuit from the zinc side to the manganese dioxide side, driven by the energy released by the chemical reaction. The battery only operates when connected; with no external wire, no electrons can travel, and the chemical reactions stop. This is why batteries have a long shelf life.

In a rechargeable battery, the chemistry is reversible: forcing current to flow backward through the battery (using a charger) rebuilds the original chemical compounds, restoring the stored energy. Lithium-ion batteries, used in laptops and electric vehicles, work on the same principle with chemistry that stores more energy per kilogram.

An electric car is conceptually simple: a large lithium-ion battery pack feeds current to an electric motor connected to the drive wheels. The motor also operates as a generator during braking, recovering some kinetic energy back into the battery. Unlike a hybrid, an electric car carries no gasoline engine at all; the entire energy supply comes from the battery, which must be recharged from the electrical grid.

Hydrogen Fuel Cells

A hydrogen fuel cell is, at its heart, just a battery — one that uses hydrogen and oxygen as its two chemical reactants. Hydrogen gas is stored in a high-pressure tank on the vehicle; oxygen comes from the air. A membrane between the two gases allows hydrogen ions (protons) and water to pass but blocks electrons. When connected to a motor, hydrogen on one side breaks apart into protons and electrons; the electrons flow through the external circuit (doing useful work), the protons pass through the membrane, and on the other side the protons, electrons, and oxygen from the air combine to form water — the only exhaust product.

The reaction starts only when the circuit is closed; with no external connection, hydrogen and oxygen simply sit in their respective chambers and nothing happens. The fuel cell can be refueled quickly by filling the hydrogen tank, unlike a battery which must be recharged over hours. The main practical challenges are producing hydrogen cleanly, storing it safely at high pressure, and building a network of hydrogen filling stations.

A hydrogen fuel cell vehicle is fundamentally an electric car: the fuel cell replaces the battery as the energy source, feeding an electric motor. The same regenerative braking and motor-generator technology applies.

Solar Energy

Solar energy can be harvested thermally or electrically. Thermal systems use mirrors or lenses to concentrate sunlight onto a small area, generating intense heat that boils water, produces high-pressure steam, and drives turbines connected to generators — the same conversion chain as a conventional power station, with sunlight replacing coal or gas as the heat source. Parabolic-trough collectors focus sunlight onto a pipe carrying fluid that heats water; a simpler version runs pipes along a roof to heat swimming pools.

A solar cell converts sunlight directly into electrical energy — no heat, no turbine. Understanding how requires quantum mechanics.

Quantum Mechanics and Neon Lights

Classical physics treats electrons as tiny charged particles orbiting nuclei in fixed circular paths. This picture fails at the atomic scale. In the quantum description, electrons behave as standing waves around the nucleus. The shape of these standing waves — the orbitals — determines the probability of finding the electron at a given location. Crucially, only certain standing-wave patterns are allowed, and each pattern corresponds to a discrete energy level. An electron cannot have just any energy; it must occupy one of these specific levels.

Light also has a dual nature: it travels and interferes as a wave, but it is absorbed and emitted in discrete packets called photons. The energy of a photon depends on its frequency (colour): blue photons carry more energy than red ones.

When a voltage is applied to a glass tube filled with neon gas, electrons in the gas absorb energy and jump to higher energy levels. These excited states are unstable; the electrons fall back to lower levels almost immediately, releasing the energy difference as a photon. Because neon’s energy levels are fixed by the laws of quantum mechanics — the same for every neon atom everywhere — that photon always has the same energy and therefore the same colour: the characteristic orange-red of neon signs. Every chemical element has a unique set of energy levels and therefore a unique emission spectrum — a fingerprint of distinct spectral lines.

Band Structure, Solar Cells, and Photoconductors

When atoms are packed into a solid, their discrete energy levels broaden into energy bands — dense clusters of closely spaced levels. The bands are separated by band gaps where no electron states exist. This band structure explains the fundamental difference between conductors and insulators.

In an insulator, the highest occupied band is completely full. For an electron to move through the material it must jump to the next empty band, but the gap is too large for ordinary electric fields to supply. No current flows.

In a conductor, the highest occupied band is only partially filled. Electrons near the top can move into empty states in the same band, flowing freely in response to an electric field. This is what distinguishes copper from glass.

A photoconductor starts as an insulator but becomes conducting when illuminated. A photon with enough energy can kick an electron from the filled valence band across the gap into the empty conduction band, simultaneously creating a mobile electron in the upper band and a mobile hole (the absence of an electron) in the lower band. Both can carry current. Remove the light and the electron falls back, the material returns to being an insulator. This is the operating principle of the photosensitive drum in a photocopier.

A semiconductor sits between insulator and conductor. By doping — introducing tiny concentrations (one in a million atoms) of impurity atoms with one more or one fewer electron than the host crystal — the band structure can be tuned. An n-type semiconductor has extra electrons in the conduction band (the “n” stands for negative charge carriers). A p-type semiconductor has extra holes in the valence band (positive charge carriers).

When p-type and n-type materials are joined, electrons from the n-side diffuse into the p-side and fill holes, leaving a region depleted of mobile charges — the depletion region — with an internal electric field pointing from n to p. This p-n junction forms a diode: electrons flow easily from n to p (conventional current from p to n), but cannot flow in the reverse direction because the internal field repels them. A diode is a one-way valve for current.

In a solar cell, the p-n junction is illuminated. Photons knock electrons into the conduction band within the depletion region. The built-in electric field immediately sweeps these electrons across to the n-side and holes to the p-side, separating the charges. Connected to an external circuit, this separation drives a current — electrical energy generated directly from light. The colour of light matters: only photons with enough energy to bridge the band gap can contribute. For silicon, this corresponds to red and shorter wavelengths.

An LED (light-emitting diode) does the reverse: electrons injected across the junction fall from the conduction band back into the valence band, releasing the energy difference as a photon. The band gap determines the colour; blue LEDs require a larger gap than red ones, which is why blue LEDs took longer to develop and require a higher forward voltage.

A simple DC power supply can be built from three components: a transformer to step down the 120 V AC from the wall to a lower voltage; a diode to remove the negative half-cycles (leaving a lumpy positive voltage); and a large capacitor to smooth the remaining ripple by charging during voltage peaks and discharging during troughs. This is the working principle of the heavy transformer-based “bricks” used for older electronics.

Rainbows and Blue Sky

A rainbow forms when sunlight enters spherical raindrops, refracts at the surface (separating colours just as a prism does, since different wavelengths slow by different amounts in water), undergoes total internal reflection off the back of the drop, and refracts again on the way out. The geometry requires the observer to have the sun directly behind them and the rain ahead. The angular spread of the exit colours means that red light reaches the eye from drops slightly higher in the sky, and blue from drops slightly lower — producing the familiar arc. A secondary (double) rainbow is produced by light that reflects twice inside the drop; it appears at a larger angle, with colours reversed.

Blue sky arises from Rayleigh scattering. Gas molecules in the atmosphere are much smaller than the wavelength of light, and — like tiny antennas — they respond more strongly to higher-frequency (shorter-wavelength) light. When sunlight passes through the atmosphere, blue and violet wavelengths are absorbed and re-emitted in random directions by the air molecules; red and orange wavelengths pass through largely unaffected. The sky glows blue because scattered blue light reaches your eyes from every direction overhead.

At sunset or sunrise, sunlight travels through a much greater thickness of atmosphere before reaching the observer. Blue and green wavelengths have been almost entirely scattered away after this long journey; only the red and orange wavelengths remain to paint the horizon.

Week 10: Rockets and Nuclear Physics

Rockets

To understand how a rocket works, you first have to think about the most fundamental principle in classical mechanics: conservation of momentum. Before a rocket fires, it is sitting still — the total momentum of the system (rocket plus propellant) is exactly zero. When the engine ignites, hot gases are accelerated out the back at tremendous speed. Because momentum must be conserved, the rocket is pushed forward with equal and opposite momentum. There is nothing mysterious about this: the rocket is simply throwing mass in one direction and recoiling in the other, exactly the same physics as a rifle recoiling when it fires a bullet. The difference is that a rifle fires one bullet, while a rocket continuously ejects propellant throughout the burn.

A rocket does not need anything to push against. Unlike a propeller-driven airplane, which grabs air and pushes backward, a rocket operates just as well — in fact, better — in the vacuum of space where there is no atmosphere to interfere. Everything the rocket needs is contained within it. This is the essential feature that makes space travel possible.

You can see this at the kitchen-table scale with a water rocket. Fill a plastic bottle partway with water, pressurize it with air, and then release it: the pressurized air forces the water out the nozzle, and the bottle shoots upward. The water is the propellant; the air is the stored energy. As the water exits rapidly downward, the rocket accelerates upward. Eventually the water runs out and only air remains — the thrust drops off and the rocket coasts. It is a satisfying demonstration precisely because it makes the momentum argument tangible.

Rocket Engines

Real rocket engines work on the same principle but with far more energetic chemistry. A solid-fuel rocket is conceptually the simplest: a casing is packed with a solid propellant — essentially a very carefully controlled explosive — and when ignited at one end it burns progressively, producing hot gas that exits through a nozzle. Solid rockets are simple, storable, and reliable. Space Shuttle solid rocket boosters and most military missiles use solid fuel. The disadvantage is that once lit, they cannot easily be throttled or turned off.

The geometry of the nozzle is critical to efficiency, and here is where some elegant fluid mechanics enters the picture. In a simple tube, gas flowing through a narrowing passage will accelerate — this is familiar from squeezing a garden hose. But for a gas travelling above the speed of sound, the physics inverts: accelerating supersonic flow requires an expanding cross-section. A De Laval nozzle exploits both regimes. The nozzle first converges (narrows), accelerating subsonic gas up to exactly the speed of sound at the narrowest point called the throat, and then diverges (expands outward) to continue accelerating the now-supersonic exhaust to very high velocities. The higher the exhaust velocity, the more momentum carried away per kilogram of propellant, and therefore the more forward thrust per kilogram burned. This is why all high-performance rocket nozzles have that characteristic bell shape.

Liquid-fuel rockets were pioneered by Robert Goddard, who in 1926 launched the world’s first liquid-fuelled rocket — a spindly device running on liquid oxygen and gasoline that flew for about 2.5 seconds and reached a height of roughly 12 metres. Laughably modest by later standards, but the concept it proved was transformative. Liquid-fuel engines pump separate tanks of fuel and oxidizer through a combustion chamber, giving much finer control over thrust. They can be throttled, restarted, and in principle designed for very high specific impulse. The Saturn V that sent astronauts to the Moon burned liquid hydrogen and liquid oxygen; modern SpaceX Merlin engines burn kerosene and liquid oxygen.

The final refinement that made orbital spaceflight practical is staging. A rocket starting from Earth must carry not only its payload but also the structural weight of the empty fuel tanks, engines, and plumbing. Once that propellant is burned, those empty components are dead weight. In a staged rocket, when a stage exhausts its fuel it is simply discarded, and the next stage — which was sitting atop the first, already moving at high speed — ignites with a much lighter vehicle to accelerate. The Saturn V had three stages: the first stage did the hardest work of lifting out of the dense lower atmosphere, the second stage boosted the vehicle to near-orbital speed, and the third stage provided the final push to the Moon.

Nuclear Physics

The Nucleus: Protons, Neutrons, and the Strong Force

The nucleus of an atom is composed of two types of particles: protons, each carrying a single positive electric charge, and neutrons, which are electrically neutral. The number of protons defines which element an atom is — one proton is hydrogen, six is carbon, 92 is uranium. The total number of protons plus neutrons is called the mass number. Atoms of the same element that differ in their neutron count are called isotopes; carbon-12 has six protons and six neutrons, while carbon-14 has six protons and eight neutrons. Most of chemistry depends only on the electron cloud, not the nucleus, so isotopes of the same element behave almost identically chemically. This is enormously useful — it means you can enrich uranium-235 among natural uranium by chemical means only with difficulty (they are nearly chemically identical), which is why uranium enrichment is so technologically challenging.

Here is a puzzle: why does the nucleus hold together at all? Protons are all positively charged, and like charges repel each other. Pack dozens of protons into a space the size of 10 to the power of minus 15 metres and the electrostatic repulsion is enormous. The answer is the strong nuclear force, one of the four fundamental forces of nature. Unlike electromagnetism, which operates over long ranges, the strong force is extremely short-ranged — it acts only when particles are essentially touching, within about one femtometre of each other. But at that range it is overwhelmingly stronger than electromagnetism, easily overcoming the repulsive force between protons and binding nucleons tightly together. Neutrons participate in the strong force but not electrostatic repulsion, which is why heavy stable nuclei have more neutrons than protons: extra neutrons add binding without adding repulsive charge.

E = mc² and Nuclear Energy

Einstein’s 1905 special relativity brought with it one of the most famous and consequential equations in all of science: \(E = mc^2\). This says that mass and energy are equivalent; a small amount of mass corresponds to an enormous amount of energy, because c, the speed of light, is about \(3 \times 10^8\) metres per second, making \(c^2\) approximately \(9 \times 10^{16}\) joules per kilogram. Converting even one gram of mass entirely into energy would release \(9 \times 10^{13}\) joules — roughly the energy of a large nuclear bomb.

Nuclear binding energy is where this matters practically. When protons and neutrons combine to form a nucleus, the resulting nucleus has slightly less mass than the sum of its parts. That missing mass has been converted into binding energy that holds the nucleus together. If you can rearrange nucleons so that the products have a lower total mass than the reactants, the difference appears as released energy. Two processes can do this: fission and fusion.

Fission is splitting a heavy nucleus into two lighter ones. Uranium-235, when struck by a neutron, can split into lighter nuclei such as barium and krypton, releasing enormous energy plus two or three additional neutrons. The key insight is that the fission products are, per nucleon, more tightly bound than the original uranium, so the total mass decreases slightly and energy is released.

Fusion is combining two light nuclei into a heavier one. Combining two hydrogen isotopes — specifically deuterium (hydrogen-2, one proton plus one neutron) and tritium (hydrogen-3, one proton plus two neutrons) — produces helium-4 and a free neutron, releasing even more energy per unit mass than fission. This is the reaction that powers the Sun and hydrogen bombs. The challenge is that to bring two positively-charged nuclei close enough to fuse, you must overcome the electrostatic repulsion, which requires temperatures of tens of millions of degrees. The Sun achieves this through gravitational compression; hydrogen bombs achieve it momentarily using a fission bomb as a trigger.

Half-life is the characteristic time over which a radioactive sample loses half its activity. After one half-life, half of the original radioactive nuclei have decayed; after two half-lives, one quarter remain; after ten half-lives, less than one-thousandth remain. Half-lives range from fractions of a second to billions of years depending on the isotope. Cobalt-60, used in radiation therapy, has a half-life of about 5.3 years. Carbon-14, used for radiocarbon dating, has a half-life of about 5,730 years. Uranium-238, the most common isotope of uranium, has a half-life of 4.5 billion years — comparable to the age of the Earth.

Heavy water is water in which the ordinary hydrogen atoms are replaced by deuterium. It has slightly different physical properties from ordinary water and is valuable in nuclear reactors as a moderator (see below).

Chain Reactions and Critical Mass

The power of fission is unleashed through a chain reaction. When a uranium-235 nucleus fissions, it releases two or three neutrons. If at least one of those neutrons strikes another fissionable nucleus and causes it to fission as well, that second fission releases more neutrons, which trigger more fissions, and so on. The reaction is self-sustaining and, if unchecked, exponential.

A useful analogy is a mousetrap demonstration. Arrange a grid of loaded mousetraps, each with two ping-pong balls balanced on the release plate. Drop a single ping-pong ball onto one trap: it springs, launching its two balls, which each land on another trap, which launches four more balls, and within a fraction of a second the whole grid is erupting in a cascade. The ping-pong balls are the neutrons; the mousetraps are the uranium nuclei.

Whether a chain reaction is self-sustaining depends on geometry and density. A tiny piece of uranium will let most neutrons escape through the surface before hitting another nucleus — the reaction fizzles. Above a certain size called the critical mass, enough neutrons are captured internally that the chain reaction sustains itself. For uranium-235, the critical mass for a bare sphere is about 52 kilograms with a diameter of roughly 17 centimetres. For plutonium-239, the critical mass is about 10 kilograms. These numbers can be dramatically reduced by surrounding the fissile material with a neutron reflector such as beryllium or natural uranium, which bounces escaping neutrons back into the core.

Nuclear Weapons

The gun-type fission bomb, used at Hiroshima in August 1945, is conceptually the simplest nuclear weapon. Two subcritical masses of highly enriched uranium-235 are kept separate inside a tube. When detonated, a conventional explosive charge fires one piece (the bullet) down the tube at high speed into the other (the target), instantly assembling a supercritical mass. The rapid chain reaction releases enormous energy. The device worked reliably but was inefficient — much of the uranium fissioned only partially before the explosion blew the material apart — and it required extremely large amounts of highly enriched uranium.

The implosion-type bomb, used at Nagasaki and used for all plutonium devices, addresses the efficiency problem. A sphere of fissile material — usually plutonium — is surrounded by precisely shaped conventional explosive lenses. All the explosives are detonated simultaneously, creating a perfectly symmetrical inward shock wave that compresses the plutonium to higher density, greatly reducing its critical mass and initiating a more complete chain reaction. The engineering challenge is extraordinary: the implosion must be exactly symmetric; any deviation creates a jet rather than an inward wave, and the bomb fizzles.

Plutonium must be produced artificially in a reactor; it cannot be enriched from naturally occurring ore. Uranium-235, by contrast, occurs naturally at 0.7% abundance in natural uranium, which is mostly uranium-238. Separating U-235 from U-238 is called enrichment, and because the two isotopes are chemically identical, it must be done by physical methods. Gaseous diffusion (lighter U-235F₆ diffuses slightly faster than U-238F₆) and gas centrifuges (which exploit the tiny mass difference at high rotational speed) are the two principal techniques. Weapons-grade uranium requires enrichment above 90% U-235; reactor fuel requires only 3–5%.

Thermonuclear weapons — hydrogen bombs — use a fission bomb as the trigger to ignite a fusion reaction, which can then drive further fission. In the Teller-Ulam design the sequence is fission → fusion → fission, often called the three F’s. A fission primary releases x-rays that compress and ignite a secondary containing deuterium-tritium fusion fuel and usually a uranium-238 or plutonium jacket around it. The fusion reaction releases fast neutrons energetic enough to fission even U-238, which does not fission in a conventional reactor. This final fission stage is where most of the explosive yield comes from, and it also produces most of the radioactive fallout. There is, in principle, no upper limit to the yield of a thermonuclear weapon — you just add more fusion stages.

Nuclear Reactors

A nuclear reactor produces a controlled chain reaction rather than an explosive one. The design challenge is to keep exactly one neutron from each fission event going on to trigger another fission — a condition called criticality — and to sustain this steadily. Two tools control the reaction: moderators and control rods.

A moderator slows down the fast neutrons produced by fission. This matters because uranium-235 fissions much more readily when struck by slow (“thermal”) neutrons than fast ones. Water, heavy water, and graphite are the three common moderators; ordinary water also absorbs some neutrons, which slightly reduces efficiency.

Control rods are made of materials like boron or cadmium that absorb neutrons very readily. Inserting the rods deeper into the reactor absorbs more neutrons and slows the reaction; withdrawing them allows more neutrons to propagate and speeds it up. Fully inserting all control rods shuts the reactor down entirely.

A boiling water reactor (BWR) passes the reactor coolant (ordinary water) directly through the reactor core, where it boils. The steam then drives a turbine to generate electricity. A pressurized water reactor (PWR) keeps the coolant under enough pressure that it cannot boil even at 300°C or more; this primary loop transfers heat through a heat exchanger to a secondary loop where steam is generated and drives the turbine. The two-loop design prevents the turbine from ever being exposed to radioactive coolant. Most commercial reactors worldwide are PWRs.

Canada’s CANDU (Canada Deuterium Uranium) reactor uses heavy water as both moderator and coolant, and as a result it can run on natural uranium — no enrichment required. This has geopolitical advantages. The reactor can also be refuelled while at full power (online refuelling), which is excellent for maintaining continuous generation.

Breeder reactors are designed to produce more fissile material than they consume. A fast reactor (using unmoderated, fast neutrons) with a blanket of U-238 or thorium around the core can convert those materials into plutonium-239 or uranium-233 through neutron capture. In principle, the world’s vast reserves of otherwise non-fissile U-238 could be used as fuel in a breeder economy. In practice, breeders have proven technically difficult and expensive.

Reactor Accidents

Nuclear power accidents share a common feature: none of them have been nuclear explosions. A reactor cannot explode like a bomb — the fuel is not enriched nearly highly enough, and the geometry is all wrong for an explosive supercritical assembly. What can happen is a thermal or chemical explosion as a consequence of overheating, combined in some cases with the release of radioactive material. Understanding the difference matters enormously for how to think about nuclear safety.

The Chalk River accident in 1952 was the world’s first significant reactor accident. A partial meltdown of the NRX reactor in Ontario occurred following a series of operational errors that led to a power surge and ruptured fuel elements. The contaminated coolant flooded the reactor basement. A young US Naval officer named Jimmy Carter, later the 39th President of the United States, led a team of military personnel in the cleanup. He was among those exposed to radiation during the operation.

Windscale (1957) involved a graphite-moderated British plutonium-production reactor. The Wigner energy effect — energy stored in the graphite crystal lattice due to radiation damage — was being deliberately released by controlled heating when the temperature was misread and the graphite caught fire. The fire burned for three days and released radioactive iodine-131 and other isotopes into the atmosphere. Milk from farms downwind was banned for weeks. Windscale was not a nuclear explosion but a graphite fire, yet it caused real environmental contamination.

Three Mile Island (1979) was a partial meltdown in a PWR in Pennsylvania. A series of equipment failures and operator errors caused the reactor core to overheat while coolant drained away. About half the core melted. However, the containment structure performed its function: almost none of the radioactive material escaped the building. The radiation dose to the surrounding population was small. The accident produced no deaths attributable to radiation and had negligible measurable health consequences, though it severely damaged public confidence in nuclear power in the United States and effectively halted new reactor construction there for decades.

Chernobyl (1986) was the most serious nuclear accident in history. The RBMK reactor at Chernobyl Unit 4 was graphite-moderated and had a crucial design flaw: under certain conditions (low power, low coolant flow) it had a positive void coefficient, meaning that if the coolant began to boil or was lost, the reaction would speed up rather than slow down — the opposite of what you want in a safe design. During a safety test being conducted in an unsafe manner at 1:23 a.m. on 26 April 1986, reactor power surged beyond control. The reactor produced a steam explosion followed by a further chemical explosion — not a nuclear bomb explosion, but powerful enough to blow the 1,000-tonne reactor lid off the building and ignite the graphite moderator. The burning graphite fire lofted radioactive material kilometres into the atmosphere. Two workers died immediately; about 28 emergency workers died of acute radiation sickness in the following weeks. A large exclusion zone was established around the site. The long-term health effects — primarily thyroid cancers from iodine-131 exposure, especially in children who drank contaminated milk — are real but have proven difficult to quantify precisely. Modern Western reactor designs do not have a positive void coefficient and are fundamentally safer than the RBMK.

Fukushima Daiichi (2011) resulted not from any design flaw during normal operation but from the interaction of an extraordinary natural disaster with inadequate emergency planning. A magnitude-9.0 earthquake struck off the coast of Japan, triggering a tsunami. The tsunami overtopped the sea wall at the Fukushima plant and flooded the backup diesel generators that were supposed to power the cooling systems during the reactor shutdown. Without cooling, the decay heat from the shutdown reactors caused the fuel to overheat and melt. Hydrogen gas accumulated and exploded, damaging the reactor buildings. Radioactive material was released, and roughly 150,000 people were evacuated from the surrounding area. No one died from acute radiation sickness from the Fukushima accident; the deaths associated with it were primarily from the stress of evacuation. The lesson drawn in many countries was not that nuclear power is inherently dangerous but that emergency cooling systems must be designed to remain functional even when external power is completely lost — a requirement now mandated in safety upgrades worldwide.

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Week 11: Medical Imaging

X-Rays

X-rays were discovered in 1895 by Wilhelm Röntgen, and the “X” in their name was precisely his acknowledgement that he did not know what they were — X for unknown. Working with a discharge tube covered in opaque cardboard, Röntgen noticed that a phosphorescent screen across the room was glowing even though no visible light could possibly be reaching it. When he held his hand between the tube and the screen, the shadow of his bones appeared. His wife’s hand produced the first x-ray photograph ever taken. The technology moved from laboratory curiosity to clinical use in a matter of months — an astonishing pace even today, and one that came with a price: overexposure to x-rays caused burns that took days or weeks to appear, and the radiation risks were not understood for years. Shoe-fitting fluoroscopes — machines that let you see the bones of your foot inside a shoe — were still present in some shoe shops into the early 1970s before their hazards were fully appreciated.

As electromagnetic waves, x-rays are no different in kind from radio waves or visible light; they just have a much shorter wavelength. Diagnostic x-rays have wavelengths around an angstrom (\(10^{-10}\) metres), comparable to the size of an atom. The shorter the wavelength, the higher the energy.

There are two standard ways to produce x-rays. The first is called Bremsstrahlung (German for “braking radiation”). When a fast-moving electron is decelerated — for instance, when it plunges into a metal target — it experiences rapid acceleration (change in velocity counts as acceleration), and any accelerating charge emits electromagnetic radiation. The result is a broad, continuous spectrum of x-ray photon energies. The second mechanism is characteristic radiation or x-ray fluorescence. The high-energy electron knocks a tightly bound inner-shell electron out of a target atom, and an outer-shell electron falls down to fill the vacancy, releasing a photon of precisely defined energy that depends on the atomic energy levels involved — exactly the same physics as neon light, but at x-ray rather than visible-light energies.

In practice, an x-ray tube contains a heated tungsten filament. Electrons boil off, are accelerated across a large voltage difference (about 30,000 volts for a hand, 100,000 volts for a chest), and slam into the tungsten or molybdenum anode target. The impact generates intense heat, so the target is usually a rotating anode that continuously presents a fresh surface to the beam, spreading the heat load and allowing much higher x-ray intensities without melting.

Energy in x-ray physics is typically expressed in electron volts (eV). One electron volt is the kinetic energy gained by a single electron accelerated through a potential difference of one volt. An electron accelerated through 100,000 volts acquires 100 kiloelectron volts (100 keV) of energy. These are convenient units because the accelerating voltage directly sets the maximum photon energy of the Bremsstrahlung spectrum.

Interactions of X-Rays with Matter

Understanding what x-rays do inside a body requires knowing four distinct interaction modes between x-ray photons and atoms.

Elastic scattering (Rayleigh scattering) is the same process responsible for the blue sky. An x-ray photon is absorbed and immediately re-emitted by an atom with the same energy but in a random direction. No energy is deposited; the photon simply changes direction. The practical consequence in imaging is that scattered photons create a diffuse background fog on the detector, reducing image contrast.

The photoelectric effect is the dominant interaction for diagnostic imaging and the one responsible for the contrast between bone and soft tissue. An x-ray photon is absorbed by a tightly bound inner-shell electron, which is then ejected from the atom. The photon’s energy must exceed the electron’s binding energy; any excess goes into kinetic energy of the ejected electron. The crucial fact is that the probability of photoelectric absorption scales strongly with the atomic number (number of protons) of the absorbing atom — approximately as the cube or fourth power of Z. Calcium (Z = 20), the principal mineral in bone, absorbs x-rays far more strongly than carbon, oxygen, or hydrogen (Z = 6, 8, 1) that make up most of soft tissue. This is exactly why bones appear white (few x-rays transmitted) and flesh appears grey or black (most x-rays transmitted) on a conventional radiograph. Iodine (Z = 53) is used as a contrast agent — injected into blood vessels or organs — because its high atomic number makes it absorb x-rays strongly, silhouetting structures that would otherwise be invisible.

Compton scattering dominates at higher photon energies. An incoming x-ray photon ejects an electron, but unlike the photoelectric effect, only part of the photon’s energy goes to the electron — the rest continues as a lower-energy scattered photon in a new direction. Compton scattering was instrumental in proving that photons carry momentum: conservation of momentum between the incoming photon, the outgoing electron, and the scattered photon was confirmed experimentally by Arthur Compton in 1923, earning him the Nobel Prize. At diagnostic x-ray energies, Compton scattering is secondary to the photoelectric effect; at the higher energies used in radiation therapy, it becomes the dominant energy-deposition mechanism.

Pair production occurs when a very-high-energy photon (above 1.02 million electron volts) interacts with the electric field near a nucleus and spontaneously converts into an electron–positron pair. The minimum threshold of 1.02 MeV corresponds to twice the rest-mass energy of an electron ((m_e c^2 = 0.511\ MeV), since both an electron and its antimatter equivalent, a positron, must be created simultaneously to conserve charge. Any excess photon energy beyond this threshold becomes kinetic energy of the two particles. The electron will ionize tissue as it travels; the positron, after slowing down, will encounter an ordinary electron, annihilate with it, and release two 0.511 MeV gamma rays. This is the basis of PET (positron emission tomography) scanning. Pair production is irrelevant for diagnostic imaging but important in high-energy radiation therapy.

Every interaction that ejects an electron deposits energy in tissue. That ejected electron can break a chemical bond directly, or it can ionize another molecule, creating reactive free radicals that then damage DNA or other biological molecules. This is the basis of radiation damage. For a routine chest x-ray the dose is small and the risk is negligible; nevertheless, medical radiation should always be justified by clinical need and minimized wherever possible.

X-Ray Imaging

A conventional x-ray image is a shadow. The x-ray source emits a diverging beam that passes through the patient’s body; the transmitted beam is recorded on the detector on the other side. Structures that absorb strongly — bones, metallic implants, calcifications — cast shadows and appear bright (white) on the displayed image. Soft tissue transmits more x-rays and appears grey; air-filled spaces (lungs, sinuses) transmit almost everything and appear dark (black). In the traditional negative convention used in radiology, dense structures are white and air is black.

Modern detectors have replaced photographic film in most radiology departments. Storage phosphors (photostimulable phosphor plates) absorb x-ray energy and store it in metastable electronic states; a scanning laser then releases the stored energy as visible light, which is read by a photodetector and converted to a digital image. Fully solid-state flat-panel detectors, similar in principle to the sensors in digital cameras, provide direct digital readout with no intermediate step.

Scatter from elastic scattering reduces image contrast by exposing the detector uniformly from all directions. Anti-scatter grids — a fine array of lead strips aligned so that only forward-directed photons pass through — are placed in front of the detector to absorb obliquely arriving scattered photons at the cost of some additional patient dose.

Dental x-rays exploit the same physics on a small scale. The familiar intraoral sensor (formerly a piece of film in a protective wrapper, now usually a storage phosphor plate or direct digital sensor) placed inside the mouth records a shadow image of the teeth and surrounding bone. Metal fillings, crowns, and implants absorb x-rays far more strongly than tooth enamel, which in turn absorbs more than soft gum tissue, producing a layered image of increasing brightness. A panoramic x-ray (orthopantomogram) uses a moving detector and source that rotate together around the patient’s head in a precisely coordinated arc that keeps the focal plane along the dental arch. Structures outside this plane — including anything in the sinuses — appear blurred, but the entire jaw from condyle to condyle is captured in one flat image useful for assessing wisdom teeth, jaw fractures, and bone pathology.

CT Scanning

A conventional x-ray provides a two-dimensional projection — all structures in the path of the beam are superimposed. A broken rib and the heart behind it both contribute to the image at the same location on the detector. This lack of depth perception limits diagnosis for anything not clearly outlined.

Computed tomography (CT) solves this by taking x-ray “shadow” measurements from many different angles around the patient and using a computer to reconstruct cross-sectional images. The principle is simple to grasp with a physical demonstration. If you look at your four fingers held together from one direction (end-on), you see an undifferentiated mass; you cannot tell whether there are one or four fingers. But if you rotate your hand and look at them from multiple angles, you gather complementary information that your brain assembles into a three-dimensional mental model. A CT scanner does exactly this mathematically.

The CT gantry contains an x-ray tube on one side and a large arc of detectors on the other, all rotating rapidly around the patient lying on a motorized table. As the table moves the patient slowly through the rotating gantry, a helical path of measurements is collected. Mathematical algorithms — variants of a technique called filtered back-projection — reconstruct the individual absorption at every point in a thin axial slice of the body. Stacking hundreds of such slices produces a volumetric dataset that can be displayed as any desired cross-section or rendered as a three-dimensional surface model.

The image quality achievable is remarkable: modern CT scanners can resolve individual eye muscles, distinguish sinuses from brain tissue, and identify tumours measured in millimetres. The tradeoff is radiation dose — a CT scan of the chest delivers roughly 100 to 200 times the dose of a single chest x-ray. For this reason CT scans are not ordered casually, but when the clinical question demands three-dimensional anatomical information, CT is often indispensable, and the diagnostic benefit typically far outweighs the radiation risk.

Radiation Therapy

Radiation therapy turns x-ray damage from a hazard into a tool. The goal is to deposit enough energy in a tumour to kill its cells while minimizing injury to the surrounding healthy tissue. At the energies used in therapy — typically one to twenty million electron volts, far above diagnostic x-ray energies — Compton scattering and pair production are the dominant interaction modes. Both create energetic electrons and positrons that in turn ionize DNA within tumour cells, causing double-strand breaks that trigger cell death.

A key geometric insight is that damage accumulates linearly along each beam path, but the tumour lies at the intersection of all paths. By rotating the radiation source around the patient (or by using multiple fixed beam angles converging on the target), each surrounding tissue element is irradiated from only one direction and receives a fraction of the total dose, while the tumour at the convergence point receives contributions from every direction. Modern intensity-modulated radiation therapy (IMRT) computers control the beam shape and intensity at each angle with multileaf collimators, sculpting dose distributions that conform tightly to irregular tumour shapes.

There are two sources of the high-energy photons required. The older approach uses cobalt-60, a radioactive isotope that undergoes beta decay: a neutron in the cobalt nucleus converts to a proton (emitting an electron — the beta particle — and an antineutrino), transforming cobalt-60 into nickel-60. The resulting nickel nucleus is in an excited state and promptly emits two gamma rays with energies of 1.17 and 1.33 MeV — both above the 1.02 MeV threshold for pair production. Cobalt-60 units are mechanically simple and do not require electrical power to produce radiation, which made them widely used in hospitals without heavy infrastructure. Their disadvantage is that the source must be periodically replaced (cobalt-60 has a 5.3-year half-life) and cannot be switched off.

The modern standard is the linear accelerator (linac). Rather than accelerating electrons through a single large voltage (which would require unrealizable megavolt power supplies), a linac uses a sequence of cylindrical conducting cavities driven by alternating electric fields. An electron enters the first cavity and is accelerated by the electric field; as it enters the next cavity the field has reversed (timed so the electron always sees an accelerating field), giving it another energy boost; this repeats through tens or hundreds of cavities until the electron reaches several MeV. The electron beam then strikes a tungsten target, producing Bremsstrahlung gamma rays through a process analogous to diagnostic x-ray production but at far higher energies. Alternatively, in proton therapy and carbon ion therapy, the heavy particles themselves — rather than x-rays — are directed into the tumour, depositing most of their energy in a sharp Bragg peak at the end of their range, sparing tissue beyond the tumour entirely.

Magnetic Resonance Imaging

Magnetic resonance imaging (MRI) — originally called nuclear magnetic resonance (NMR) until the word “nuclear” was judged too alarming for patients — exploits a quantum-mechanical property of protons rather than ionizing radiation. It causes no radiation damage whatsoever, and it provides exquisite soft-tissue contrast that x-rays and CT cannot match.

The proton at the heart of every hydrogen atom has a property called spin, which causes it to behave, in a loose analogy, like a tiny bar magnet with a north and south pole. When placed in a strong external magnetic field, proton spins tend to align with the field, just as a compass needle aligns with the Earth’s field. But alignment is not perfect: thermal energy at body temperature continuously disrupts the alignment. The net result is a slight statistical excess of spins pointing parallel to the external field over those pointing anti-parallel.

This aligned spin state and the anti-aligned state have slightly different energies, split by an amount that depends precisely on the strength of the local magnetic field. The energy difference corresponds to a radio-wave photon — which is why MRI uses radio frequencies rather than x-rays. To probe the spins, the MRI machine transmits a brief pulse of radio waves at exactly the right frequency (the Larmor frequency) to flip a fraction of the aligned spins into the anti-aligned state. When the pulse is turned off, those spins relax back toward their equilibrium alignment, emitting radio-frequency photons as they do. The frequency of the emitted photon encodes the local magnetic field strength that the spin experienced.

The three-dimensional imaging trick is elegant. The main superconducting magnet produces a very strong, nearly uniform field (typically 1.5 to 3 Tesla — tens of thousands of times stronger than the Earth’s magnetic field). Overlaid on this are three sets of gradient coils that add small, carefully calibrated spatial variations to the field along the x, y, and z directions. Because the Larmor frequency depends on field strength, each spatial location in the body has a unique resonance frequency. By analysing which frequencies are present in the emitted signal and at what amplitude, the scanner can reconstruct the position of every proton that responded — a three-dimensional map.

The relaxation time adds a further dimension of information. Different tissues — fat, muscle, grey matter, white matter, tumour — return their spins to equilibrium at different rates, because the relaxation rate reflects the local chemical and physical environment of the hydrogen nuclei. By varying the timing parameters of the radio pulses, the radiologist can weight the image to emphasise different types of tissue contrast. This is why MRI scans of the brain can distinguish fine anatomical structures that would be invisible on CT: the relaxation differences between grey matter, white matter, and cerebrospinal fluid are substantial.

The characteristic banging and clanging noise that anyone who has had an MRI will recognise comes from the gradient coils. The gradient coils carry pulsed electrical currents inside the intense field of the main magnet; the magnetic force on a current-carrying wire (\(F = IL \times B\) causes the coils to flex mechanically with each pulse. Modern scanner designs incorporate acoustic dampening, but the banging cannot be eliminated entirely at clinical field strengths.

MRI requires no ionising radiation and no contrast dye for most applications. It can image veins and arteries using flow-sensitive sequences (MR angiography) without injection. It can perform functional MRI (fMRI), detecting the small signal changes associated with increased blood oxygenation in active brain regions — the basis of cognitive neuroscience experiments that map which brain areas respond to particular stimuli. These capabilities, combined with its non-invasive nature, have made MRI one of the most powerful and widely used diagnostic tools in modern medicine.

Radioactive Decay: The Cloud Chamber

A cloud chamber makes individual radioactive decay events directly visible. A container is filled with a supersaturated vapour — typically ethanol — cooled from below so that it hovers just on the edge of condensing. When a charged particle from a radioactive source passes through the vapour, it ionises the molecules along its path, providing nucleation sites for tiny droplets to form. The result is a visible track, like the contrail left by a jet aircraft, tracing the exact path of the particle. Alpha particles (helium-4 nuclei) produce short, straight, thick tracks because they are massive and doubly charged; they ionise the vapour densely but quickly lose energy and stop. Beta particles (electrons) produce longer, thinner, more tortuous tracks because they are light and easily deflected. A small stick coated with lead-210 (which decays by alpha emission) placed inside the chamber produces a beautiful starburst of short, straight tracks radiating outward from the tip — each track representing a single nucleus that has just undergone fission, its escaping alpha particle leaving a momentary contrail in the mist.