AQA A level Physics: Turning Points, Study Guides, Projects, Research of Physics

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Section 12: Turning Points in Physics

AQA Paper 3, Section B

Contents

The Discovery of the Electron

This section covers the experimental evidence that led to the discovery and characterisation of the electron, from early observations of cathode rays through to Millikan’s precise determination of the electronic charge.

Cathode Rays

Discharge Tubes

When a high potential difference is applied across a glass tube containing a low-pressure gas, the tube begins to glow. The glow is brightest near the cathode (the negative electrode). The luminous emission was historically called the cathode ray, and its nature was debated for decades.

The process by which a discharge tube produces light is as follows:

1. The high potential difference ionises gas atoms, pulling electrons away from them and form- ing ion–electron pairs.
2. The positive gas ions are accelerated towards the cathode. When they collide with the cathode surface, they release additional electrons.
3. These electrons are accelerated along the tube. Because the gas pressure is low, the mean free path is long, so electrons reach high speeds before colliding with gas atoms.
4. When fast electrons collide with gas atoms, they excite them to higher energy levels. The atoms quickly de-excite, releasing photons of visible light.

The glow is brightest near the cathode because this is where positive ions and electrons recom- bine, emitting photons in the process.

Properties of Cathode Rays

J. J. Thomson showed that cathode rays:

• Have mass, which he measured.
• Carry a negative charge (deflected by electric and magnetic fields).
• Have the same properties regardless of the gas used in the discharge tube.
• Have a very large charge-to-mass ratio (e/me), much larger than for any ion.

It was soon concluded that all atoms contain these particles, which were renamed electrons.

Key Point

The fact that cathode rays had the same properties regardless of the gas used demonstrated that electrons are a universal component of all atoms.

Thermionic Emission

Definition

Thermionic emission is the process by which a metal is heated until the free (delocalised) electrons near its surface gain enough kinetic energy to overcome the attractive forces holding them in the metal and are emitted into the surrounding vacuum.

While most electrons in a metal are tightly bound within atoms, typically one or two per atom are delocalised and move throughout the crystal lattice. Heating the metal increases the average kinetic energy of these electrons; a fraction near the surface acquire enough energy to escape.

Electron Guns

An electron gun uses thermionic emission combined with an accelerating electric field to produce a narrow beam of fast electrons.

1. A cathode (heated filament) emits electrons via thermionic emission.
2. The electrons are accelerated towards a positive anode by the potential difference V between cathode and anode.
3. The anode has a small hole; electrons that pass through it form a narrow, uniform-speed beam beyond the anode.

Increasing the current through the filament raises the cathode temperature, causing more electrons to be emitted per second, increasing the beam current (number density of electrons in the beam).

Work Done on an Electron

The work done on a charged particle moving through a potential difference is:

Work Done on an Electron

W = Q ∆V For an electron accelerated through a potential difference V :

W = eV

where e = 1. 60 × 10 −^19 C is the elementary charge.

This is often measured in electron volts (eV): 1 eV is the kinetic energy gained by an electron accelerated through a p.d. of 1 V, so 1 eV = 1. 60 × 10 −^19 J.

As the electron moves from cathode to anode, its electrical potential energy is converted into kinetic energy. When it reaches the anode:

Kinetic Energy of Accelerated Electron

1 2 mev^2 = eV =⇒ v =

r 2 eV me

Specific Charge of the Electron

The specific charge of the electron is the ratio e/me. You need to know how to determine this by at least one method. Two classic approaches are described below.

Method 1: Fine Beam Tube

A fine beam tube contains a low-pressure gas and sits in a uniform magnetic field B.

1. Electrons are accelerated by an electron gun and enter the tube perpendicular to the field.

Significance of Thomson’s Results

Significance

• The specific charge was constant regardless of the gas used ⇒ electrons are a universal constituent of all atoms.
• The specific charge of the electron (≈ 1. 76 × 1011 C kg−^1 ) is about 1800 times larger than that of a hydrogen ion (≈ 9. 58 × 107 C kg−^1 ), implying the electron has the same magnitude of charge but a much smaller mass.
• Thomson proposed the plum pudding model: electrons spread uniformly through a sphere of positive charge (later disproved by Rutherford scattering).

Millikan’s Oil Drop Experiment

Principle and Apparatus

Millikan’s experiment was designed to measure the charge of an electron.

1. An atomiser sprays tiny oil droplets, which become negatively charged due to friction.
2. The droplets fall between two horizontal parallel plates that form a uniform electric field.
3. The p.d. across the plates is adjusted until a particular droplet becomes stationary: the upward electric force balances the droplet’s weight.

When stationary:

Stationary Droplet Condition

QE = mg =⇒

QV

d = mg

where Q is the charge on the droplet, V is the p.d., d is the plate separation, and m is the droplet mass.

Determining the Droplet Mass Using Stokes’ Law

The mass of the droplet is unknown, so it must be determined independently. The electric field is turned off and the droplet falls under gravity, experiencing a viscous drag force given by Stokes’ law:

Stokes’ Law

F = 6πηrvt where η is the viscosity of air, r is the radius of the droplet, and vt is its terminal velocity.

The terminal velocity vt is measured using a calibrated microscope graticule (timing the droplet over a known distance). At terminal velocity, drag equals weight:

6 πηrvt =

πr^3 ρg

Solving for the radius:

r =

r 9 ηvt 2 ρg

where ρ is the density of the oil. Once r is known, the mass is m = 43 πr^3 ρ, and Q can be found from the stationary condition.

Results and Significance

Key Result

Millikan found that the charge on every droplet was an integer multiple of 1. 60 × 10 −^19 C. This demonstrated that electric charge is quantised: it exists only in discrete packets of e = 1. 60 × 10 −^19 C, the fundamental unit of charge.

Worked Example – Millikan Calculation

An oil droplet (ρ = 900 kg m−^3 ) falls at terminal velocity vt = 5. 0 × 10 −^4 m s−^1 in air (η =

1. 8 × 10 −^5 Pa · s). It is held stationary between plates (d = 5.0 mm, V = 250 V). Find Q. Step 1: Find r:

r =

r 9 × 1. 8 × 10 −^5 × 5. 0 × 10 −^4 2 × 9. 81 × 900 ≈ 2. 1 × 10 −^6 m

Step 2: Find m: m =

π(2. 1 × 10 −^6 )^3 × 900 ≈ 3. 7 × 10 −^14 kg

Step 3: Find Q:

Q =

mgd V

3. 7 × 10 −^14 × 9. 81 × 5. 0 × 10 −^3

≈ 7. 3 × 10 −^18 C

This is 7. 3 × 10 −^18 / 1. 60 × 10 −^19 ≈ 46 e – close to an integer multiple, supporting charge quantisation.

Table 1: Comparison of Newton’s and Huygens’ theories of light.

Property Newton (Corpuscles) Huygens (Waves) Reflection Repulsive force reverses perpen- dicular velocity

Wavelets reform reflected wave- front Refraction Attractive forces speed up light in denser medium

Light slows down in denser medium Speed in dense media

Faster Slower

Diffraction Cannot explain Naturally explained

Young’s Double Slit Experiment

The Experiment

In Young’s double slit experiment, coherent monochromatic light is directed at two narrow, closely spaced slits. Each slit acts as a coherent point source, and the waves from the two slits overlap to produce an interference pattern of alternating bright and dark fringes on a screen.

• Bright fringes (constructive interference): waves arrive in phase; path difference = nλ (where n is an integer).
• Dark fringes (destructive interference): waves arrive completely out of phase; path dif- ference = (n + 12 )λ.

Significance

Why Young’s Experiment Matters

• If Newton’s corpuscular theory were correct, there would be only two bright fringes (one behind each slit) and no interference pattern.
• The observation of diffraction and interference – both wave properties – provided strong evidence for Huygens’ wave model.
• Even after Young’s experiment, the wave theory was not immediately accepted be- cause of Newton’s towering reputation.
• Final acceptance came when the speed of light was measured in water and found to be slower than in air, directly contradicting the corpuscular theory.

Electromagnetic Waves

Nature of EM Waves

Electromagnetic (EM) waves consist of oscillating electric and magnetic fields that:

• Travel in phase with each other.
• Are perpendicular to each other.
• Are both perpendicular to the direction of wave propagation (transverse wave).

Maxwell’s Prediction

James Clerk Maxwell predicted the existence of EM waves from theory alone. He derived a formula for their speed in vacuum:

Maxwell’s Speed of Light

c =

μ 0 ε 0 where:

• μ 0 = permeability of free space ( 4 πe − 7 H m−^1 ) – relates the magnetic flux density produced by a wire to the current.
• ε 0 = permittivity of free space ( 8. 85 × 10 −^12 F m−^1 ) – relates the electric field strength to the charge producing the field.

Substituting the known values:

c =

4 π × 10 −^7 × 8. 85 × 10 −^12

≈ 3. 0 × 108 m s−^1

This matched the measured speed of light, providing strong evidence that light is an electromag- netic wave.

Hertz’s Discovery of Radio Waves

Heinrich Hertz provided experimental confirmation of Maxwell’s predictions:

1. Hertz used an apparatus that produced high-voltage sparks jumping across an air gap, which generated radio waves.
2. The waves were detected using:
   • A dipole receiver (parallel charged plates) – detects the wave’s electric field.
   • A loop of wire with a gap – detects the wave’s magnetic field (the changing flux induces an EMF, causing a spark across the gap).
3. A metal sheet placed in front of the apparatus reflected the waves, creating stationary waves. The distance between adjacent nodes gave the wavelength.
4. The calculated wave speed matched Maxwell’s predicted value for c.
5. Rotating the detector by 90 ◦^ caused the signal to vary from maximum to zero, demonstrating the waves were polarised – a property of transverse waves.

Fizeau’s Measurement of the Speed of Light

Armand Fizeau measured the speed of light using a toothed wheel and a distant mirror:

1. A pulsed beam passes through a gap between teeth in a rotating wheel, travels to a distant mirror, and reflects back.
2. At slow rotation speeds, the reflected light returns through the same gap.
3. The speed is increased until the light is blocked by the adjacent tooth – the light is no longer visible.
4. If the speed is doubled, the light returns through the next gap and becomes visible again.

For a wheel with n teeth and n gaps, a tooth replaces a gap after (^21) n of a revolution. The time for this is (^2) nf^1 (where f is the rotation frequency). If the mirror is at distance d:

1. A single photon transfers all its energy to a single electron (one-to-one interaction). If hf < ϕ (the work function), the electron cannot escape ⇒ threshold frequency exists: f 0 = ϕ/h.
2. Energy transfer is instantaneous – no gradual build-up.
3. Intensity = number of photons per second. More photons ⇒ more photoelectrons, but each photon has the same energy ⇒ Kmax unchanged.
4. Electrons deeper in the metal lose energy through collisions while escaping ⇒ range of kinetic energies.

Einstein’s Photoelectric Equation

hf = ϕ + Ek(max) where ϕ is the work function (minimum energy to remove a surface electron) and Ek(max) is the maximum kinetic energy of the photoelectrons.

Stopping Potential

The stopping potential Vs is the p.d. needed to stop the most energetic photoelectrons:

Ek(max) = eVs

Substituting into Einstein’s equation:

hf = ϕ + eVs =⇒ Vs = h e

f − ϕ e

A graph of Vs against f gives a straight line with:

• Gradient = h/e
• y-intercept = −ϕ/e
• x-intercept = f 0 (the threshold frequency)

This was experimentally confirmed, validating Einstein’s photon model.

Table 2: Wave theory vs. photon model predictions for the photoelectric effect.

Observation Classical Wave Theory Photon Model Threshold frequency No threshold; any f works at high intensity

Yes: hf ≥ ϕ needed

Delay in emission Possible delay None (instantaneous) Kmax vs f Should increase with inten- sity

Linear in f , slope = h

Current vs intensity Increases with intensity Increases with intensity (for f > f 0 )

Wave–Particle Duality and the de Broglie Hypothesis

The de Broglie Hypothesis

Louis de Broglie (1924) proposed that if light (a wave) can behave like a particle, then all particles should also exhibit wave-like behaviour. The wavelength of any particle is given by:

de Broglie Wavelength

λ = h mv

h p where h is the Planck constant, m is mass, v is velocity, and p = mv is momentum.

Electron Diffraction

Electron diffraction provided direct experimental evidence for the de Broglie hypothesis:

1. Electrons are accelerated by an electron gun through a vacuum tube towards a thin poly- crystalline target (or crystal lattice).
2. The electrons interact with the small gaps between atoms in the crystal (which act as a diffraction grating).
3. A diffraction pattern of concentric rings appears on a fluorescent screen behind the crystal, just as would be expected for waves.

For electrons accelerated through p.d. V , combining 12 mev^2 = eV with p = mev:

de Broglie Wavelength for Accelerated Electrons

λ =

h √ 2 meeV This is valid for non-relativistic speeds (v ≪ c).

Effect of Accelerating Voltage on Diffraction

• Increasing V ⇒ v increases ⇒ λ decreases ⇒ diffraction rings move inwards (closer together).
• Decreasing V ⇒ v decreases ⇒ λ increases ⇒ diffraction rings move outwards (further apart).

This behaviour follows wave theory (fringe spacing increases with wavelength) and provides strong evidence for the de Broglie hypothesis.

Condition for Diffraction For significant diffraction to occur, the gap size (or obstacle size) must be approximately equal to the de Broglie wavelength. Atomic spacings (∼ 0 .1 nm) are of the right order for electron diffraction.

Electron Microscopes

Resolving Power

The resolving power of a microscope is its ability to distinguish between two closely spaced structures. Because the wavelength of an electron beam is much smaller than that of visible

STM vs TEM

• TEM images the internal structure of thin samples (transmitted electrons).
• STM images the surface topography at atomic resolution (tunnelling current).

Special Relativity

This section covers the experimental motivation for special relativity (the Michelson–Morley exper- iment), Einstein’s postulates, and the three major consequences: time dilation, length contraction, and mass–energy equivalence.

The Michelson–Morley Experiment

The Luminiferous Aether

In the 19th century, scientists believed in absolute motion: all objects move relative to a substance called the luminiferous aether, which was thought to permeate all of space and serve as the medium through which light waves propagated.

The Interferometer

Michelson and Morley (1887) designed an interferometer to detect the Earth’s motion through the aether:

1. A light beam is split by a partially reflective surface (beam splitter) into two perpen- dicular beams.
2. Each beam travels to a mirror placed at equal distances from the splitter and reflects back.
3. A compensating glass plate ensures both beams pass through the same total thickness of glass.
4. The beams recombine at a detector, producing an interference pattern.

Expected vs. Observed Result

If the aether existed:

• The beam travelling parallel to the Earth’s motion through the aether would be affected (slowed in one direction, sped up in the other), taking longer overall than the perpendicular beam.
• Rotating the apparatus by 90 ◦^ should cause a shift in the interference pattern as the roles of the two beams are swapped.

The Null Result No fringe shift was observed, even after repeated measurements at different times of year (when the Earth’s velocity through the aether would differ). This led to two key conclusions:

1. The luminiferous aether does not exist (or is dragged along by the Earth – later ruled out).
2. The speed of light is invariant in free space: it is independent of the motion of the source or the observer.

Einstein’s Theory of Special Relativity

Inertial Frames of Reference

An inertial frame of reference is one that is not accelerating – it moves at a constant velocity relative to other inertial frames. A frame that is accelerating or rotating is not an inertial frame. Newton’s first law holds in all inertial frames.

Worked Example – Time Dilation

Lucy is in a spaceship travelling at 0. 9 c and measures 1 hour on her clock. How much time passes on Earth? Lucy is the stationary observer (at rest relative to her clock), so t 0 = 1 hour.

t =

r 1 − (0. 9 c)^2 c^2

≈ 2. 3 hours

Evidence: Muon Decay

Muon decay provides compelling experimental evidence for time dilation.

Muons are unstable subatomic particles created when cosmic rays strike the upper atmosphere. They have a rest-frame half-life of approximately 1. 5 μs (or mean lifetime ∼ 2. 2 μs).

Experimental setup: Two detectors are placed at different altitudes to measure the muon count rate. The distance d between them and the muon speed v are measured.

Without special relativity (classical calculation):

Using the experimental data (d = 2 km, v = 0. 996 c, initial count rate = 100 s−^1 ):

t =

d v

2 × 103

0. 996 × 3 × 108

= 6. 69 × 10 −^6 s

Number of half-lives = 6.^69 ×^10 − 6

1. 5 × 10 −^6 = 4.^46.

Expected count rate = 100 ×

2

≈ 4 .5 s−^1

This is far lower than the observed count rate of 80 s−^1.

With time dilation:

The proper time experienced by the muons is shorter:

t 0 = t

r 1 − v^2 c^2

= 6. 69 × 10 −^6 ×

p 1 − 0. 9962 ≈ 6. 0 × 10 −^7 s

Number of half-lives = 6.^0 ×^10 − 7

1. 5 × 10 −^6 = 0.^4.

Expected count rate = 100 ×

2

≈ 76 s−^1

This is very close to the observed value of 80 s−^1 , providing strong evidence for time dilation.

Length Contraction

Proper Length

Length contraction is another consequence of special relativity. The proper length (l 0 ) of an object is its length measured by an observer at rest relative to the object.

An external observer who sees the object moving at speed v measures a shorter length l:

Length Contraction

l = l 0

r 1 − v^2 c^2

l 0 γ

Important

• Length contraction only affects the dimension parallel to the direction of motion. Width and height perpendicular to the motion are unchanged.
• Never apply both time dilation and length contraction simultaneously in the same calculation – they are interlinked consequences of the same effect viewed from different frames.

Worked Example – Length Contraction

A spaceship has proper length l 0 = 120 m and flies past an observer at v = 0. 80 c. What length does the observer measure?

l = 120

p 1 − 0. 802 = 120

0 .36 = 120 × 0 .60 = 72 m

Muon Decay and Length Contraction

The muon experiment can also be analysed using length contraction (instead of time dilation). From the muon’s frame of reference, the distance between the detectors is contracted:

Using the same data (l 0 = 2 km, v = 0. 996 c):

l = 2000

p 1 − 0. 9962 ≈ 180 m

Time to travel this contracted distance:

t =

0. 996 × 3 × 108

= 6. 0 × 10 −^7 s

Number of half-lives = 6.^0 ×^10 − 7

1. 5 × 10 −^6 = 0.^4.

Expected count rate = 100 ×

2

≈ 76 s−^1

This gives the same answer as the time dilation approach – both lead to consistent predictions, as expected from special relativity.

Mass and Energy

Mass–Energy Equivalence

Einstein showed that mass and energy are interchangeable:

Mass–Energy Equivalence

E = mc^2 Even a stationary object has an associated rest energy:

E 0 = m 0 c^2