Turning points

How do CRTs work?

A PD is applied across a discharge tube with a low pressure gas in it

This will separate out the electrons from gas atoms to form ions and electrons

Positive ions → cathode → more electrons released

Electrons → anode

Electrons released collide with atomic electrons to excite then deexcite, releasing photons

What properties did Thompson show cathode particles had?

Mass

Negative charge

Large charge to mass ratio (specific charge)

Why is the glow of a CRT brightest at the cathode?

Gas ions and electron can recombine to emit photons

Thermionic emission

When a metal is heated in order for the free electrons on its surface to gain enough energy to be emitted

How do electron guns work?

1) A cathode is heated until electrons are emitted (thermionic emission)

2) There is a PD across the electrodes so the electrons are accelerated towards the anode

3) The anode has a hole in it which allows the electrons to pass through

Work done and kinetic energy of an electron emitted from an electron gun

W = eV = ½ mv²

Fine beam tube method for specific charge

1) Electrons are accelerated using an electron gun and enter the tube perpendicular to the direction of the B-field

2) The magnetic force acts perpendicular to the direction of the electrons’ motion causing circular motion

3) As the electrons move the collide with gas atoms causing excitation and deexcitation, therefore the radius of the electrons’ path can be measured

How can we find the equation for specific charge using a fine beam tube (equation)?

m_ev²/r = Bev

½ m_ev² = eV, v² = 2eV/m_e

m_e(2eV/m_e)^1/2/r = Be

m_e2V/r² = B²e

e/m_e = 2V/B²r²

Thomson’s crossed fields method for specific charge

1) Electrons are accelerated with an electron gun and enter the apparatus perpendicular to the direction of both fields

2) The crossed fields act like a velocity selector

3) The fields are adjusted until the electron passes through the fields undeflected

Equations for Thomson’s crossed fields

Bev = Ve/d, v = V/Bd

½ m_ev² = eV_a

V²/B²d² = 2eV_a/m_e

e/m_e = V²/2B²d²V_a

Purpose of the equipment in this image

PD - Provide an accelerating electric field for the oil drops

Cover - For the oil droplets to be sprayed onto

Oil spray - To negatively charge the droplets due to friction

Microscope - To record when the force due to the electric field is equal to the weight of the drop (so it is stationary in the air)

Expression for weight in Millikan’s oil experiment

mg = QV/d

Why is the PD removed from the electric field in Millikan’s experiment

As the mass is unknown, it needs to be found. By removing the PD, the oil drop will start to fall. When it reaches terminal velocity, the weight will be equal to the viscous drag force, which can be rearranged to find the radius of the oil drop

Stokes’ law for viscous drag force

F = 6πηrv

How is the speed of a falling oil drop measured?

A microscope with a calibrated graticule is used to measure the distance a drop falls in a certain time, which is then used with v = s/t

What did Millikan’s oil drop experiment show?

All the charges of the oil drops were integer multiples of 1.6 × 10^-19 C

Showing that charge is quantised (exists in discrete packets)

Corpuscular theory of light

Newton theorised that light was made up of tiny particles

Newton’s explanation of reflection

The corpuscles collide with the surface and a repulsive force pushes them back, causing their component of velocity perpendicular to the surface to change direction

Newton’s explanation of refraction

As the corpuscules approach a denser medium, short range forces of attraction cause their velocity perpendicular to the surface to increase

This means the light will bend towards the normal

What did Newton say about the speed of light in a denser medium?

It was faster

Huygen’s principle

Light is made up of waves

Where every point on a wavefront is a point source to secondary wavelets, which then spread out to form the next wavefront

How can reflection be explained with Huygen’s theory?

As the whole wavefront will not reach the surface at the same time, wavelets spread away from the surface once they reach it and rejoin with others

How can refraction be explained using Huygen’s theory?

It was assumed that light travels slower in a more dense medium (and therefore as c = fλ and f is constant, λ decreases), so it would then bend towards the normal because the wavelet that arrives at the surface first would have more time to propagate, but would propagate slower

Why was Newton’s theory originally prefered to Huygen’s?

Newton had a much higher reputation

Diffraction had not yet been observed (which would disprove Newton’s theory)

The speed of light had not been measured (in different mediums)

The existence of sharp-edged shadows

Why was Young’s double slit experiment important for the theory of light?

It showed experimental evidence for diffraction, which did not fit with the corpuscular theory of light

It also showed light interference, which is a wave property

Where does constructive interference occur in Young’s double slit experiment?

Where the path difference is nλ

Where does destructive interference occur in Young’s double slit experiment?

Where the path difference is (n + 1/2)λ

How are EM waves formed?

Of perpendicular electric and magnetic fields oscillating as transverse waves

Maxwell’s equation for the speed of EM waves

c = 1/sqrt(μ₀ε₀)

What is μ₀?

The permeability of free space

The ability of free space to support the formation of magnetic fields

What is ε₀?

The permittivity of free space

The ability of free space to support the formation of electric fields

How did Hertz’s first experiment show EM waves existed?

The dipole receiver sparked, showing that there was an induced current in it due to a B-field and that sparks were created between the capacitors, showing the electric field

How could the wavelength of EM waves be calculated?

Place a metal sheet in front of a transmitter to reflect the waves and form stationary waves. Then use a dipole receiver to find two consecutive nodes, and the wavelength will be double the length between the nodes

How did Hertz’s experiment show that EM waves are polarised?

The dipole receiver sparked more at a certain angle, and had a certain angle where no sparks were produced, meaning that the wave oscillated in a specific direction

How did Fizeau’s experiment work?

A light source was reflected by a glass plate through a rotating, toothed disk to hit a mirror at the other end, where it would then bounce back through the disk where an observer could see through the glass plate.

Formula for what fraction of a revolution of Fizeau’s disk would cause a tooth to replace a gap

1/2n (n = number of teeth and gaps)

How did Fizeau derive the speed of light from his experiment?

rev% = 1/2n

t_rev = 1/f

t_{tooth replaces gap} = 1/2nf

s = d/t = 2d/1/2nf

c = 4dnf

(where d is distance from disk to mirror)

What is a black body?

A hypothetical object that is a perfect emitter and absorber (emits and absorbs all wavelengths of radiation)

Ultraviolet catastrophe

Where the classical wave model of light caused a prediction for a small wavelength of light to emit infinite energy

What model caused the UV catastrophe?

The Rayleigh-Jean equation

Wien’s displacement law

λ_max = b/T

How did Planck’s quanta theory solve the UV catastrophe?

The experimental radiation curve could be explained in terms of individual particles. At the peak, each individual photon has the entirety of the predicted energy emission. Past that, each photon has more energy than that, so the probability of emission drops (as the system is not hot enough to radiate the photons).

Why could the photoelectric effect not be explained by wave theory?

Wave theory suggests that any frequency of light should be able to cause photoelectric emission - hence the work function should not exist

The photoelectric effect is immediate, it does not require any time for energy to build up

Increasing the intensity of light does not increase the KE_max, instead it increases the current

Photoelectrons have a range of kinetic energies

How did Einstein’s photon theory explain photoelectricity

Photons have to transfer all of their energy in a collision with an atomic electron, which explains the work function

The photon energy is immediately transferred, which explains the immediate emission of photoelectrons

Intensity is the number of photons emitted per second, so would increase the number of collisions (1:1 ratio)

All electrons receive the same energy, but some need to use more KE to escape the material

What is the stopping potential?

The minimum potential required that is applied across the metal surface that stops the photoelectrons with the highest KE

V_s = KE_max/e

Einstein’s photoelectric equation in terms of stopping potential

V_s = h/e f - ϕ/e

de Broglie hypothesis

All particles have a wave like nature and a particle like nature

de Broglie wavelength

λ = h/mv = h/p

What experiment provided evidence for the de Broglie hypothesis?

Electron diffraction

How is electron diffraction carried out?

Electrons are accelerated by an electron gun (1/2mv² = eV) towards a crystal lattice, so that the electrons’ wavelength (λ = h/sqrt(2meV)) is similar to the distance between the gaps in the lattice. A diffraction pattern of concentric circles on a screen is then formed

How did electron diffraction provide evidence for the de Broglie hypothesis?

If electrons are accelerated more, they will have a larger momentum, and therefore a smaller de Broglie wavelength. It is then observed that the fringe spacing is decreased. The converse is also true (less acceleration → smaller momentum, larger de Broglie wavelength and fringe spacing increases)

Resolving power of a microscope

The ability of a microscope to distinguish structures which are close to each other.

What happens to the resolving power of an electron microscope when the electron wavelength decreases?

It increases

TEM

Transmission Electron Microscope

What lenses does a TEM have?

Condenser

Objective

Projector

What does a condenser lens do?

It deflects the electrons from the electron gun so that they form a wide beam, which is then directed at the sample

What does the objective lens do?

This is the lens which is directly below the sample

It will form an image of the sample

What does the projector lens do?

This lens will magnify the image made by the objective lens and project it onto the fluorescent screen

What is the resolving power of a TEM limited by?

Sample thickness

Electrons travelling with a range of speeds

What is the voltage of the electron gun needed to produce an electron wavelength similar to the diameter of an atom (0.1 nm) for a TEM

150 V

STM

Scanning Tunnelling Microscope

How does an STM work?

A very fine tipped probe is moved close to the surface of a sample with a small PD applied across the probe tip and the sample

When the tip is close enough, some electrons in the sample (due to their wave function overlapping with the end of the tip) can tunnel through the gap into the tip of the probe, which forms a current which’s magnitude is proportional to the distance from the probe tip to the surface of the sample.

How does quantum tunnelling work

Quantum tunnelling allows particles to overcome potential barriers that they should not have enough energy to overcome

This occurs due to the wave function of particles, which says that particles are represented as a ‘cloud’ of probability of where they can lie

If another object overlaps with the edge of the wave function of a particle in another object, there is a small chance that the particle is found in the other object

The two modes of an STM

Constant height - current varies so can be used to determine the distance

Constant current - distance adjusted and recorded so the current is constant

What is Ether?

Ether was a hypothetical substance which scientists used to believe permeated the entire universe which was the hypothetical medium for light

What was the purpose of an interferometer?

To measure the absolute speed of the Earth through the ether

What was the interferometer made from?

Light source

3 mirrors

A beam splitter (+ a compensating plate)

The mirrors and light source were arranged such that they form a square, so the split beams of light are reflected back on themselves

Why did Michelson and Morley use an interferometer to try and prove the existence of the ether?

They believed that the speed of the light beam moving parallel to the motion of the Earth would be affected, so an interference pattern would be formed that would shift if the interferometer was rotated 90 degrees

3 conclusions drawn from the Michelson-Morley experiment

1) The ether does not exist

2) Or the Earth drags the ether along with it as it moves

3) The speed of light is invariant in free space

Inertial reference frame

Frames which move at a constant velocity relative to each other

Einstein’s special relativity postulates

1) The speed of light in free space is invariant

2) The laws of physics have the same form in all inertial frames

Time dilation

An external observer will always observe a larger time than an observer moving at a relativistic speed

Expression for speed an object moves with and its experienced time

v ∝ 1/t (Inverse proportionality)

Proper time

The time experienced by the stationary observer (within the inertial reference frame) (t₀ < t)

Lorentz factor

(γ = ) (1-v²/c²)^-1/2

How to derive proof of time dilation

Light clock - find time taken

Light clock on a moving train of velocity v

Use trig for the distances travelled by the light on the train

Rearrange for t and factorise out

What phenomenon provides evidence of time dilation and why?

Muon decay

The muons travelling between two detectors have a half life such that very few should survive the distance, but many more than predicted do

Therefore the muons at a very high speed (0.996c) must experience time slower than us

Length contraction

Objects at a relativistic speed seem shrunken for external observers

Proper length

The length measured by an observer who is at rest relative to the object

Relativistic mass

The mass observed for a body which has had energy added to it, therefore increasing its mass

Energy expression of a relativistic mass

E_T = E₀ + E_k

How to work out the total mass of a relativistic object

E_T = m₀c²*γ

Bertozzi’s experiment

Particle accelerator which accelerates groups of electrons at an aluminium plate through a detector which could determine the number of electrons emitted.

How to calculate the kinetic energy of one electron from Bertozzi’s experiment

When the electrons collide with the target all of their kinetic energy is converted to thermal energy

E_T = mcΔθ

E_{one electron} = mcΔθ/n

Why can an object never reach the speed of light?

At the speed of light, relativity says that the object will have an infinite kinetic energy, which is impossible

The Lorentz factor will also be equal to 0, so when working out the energy it will give an expression divided by 0, which is impossible