2 hours60 marks

Level 1 - Physics Paper 2

Electromagnetism, relativity, and quantum mechanics.

Instructions

Attempt all questions.
All main questions carry the same number of marks.
Show enough working to make the model, assumptions, and sign conventions clear.
Begin each main question on a new page when working on paper.

Constants

Gravitational acceleration\( g=9.81\,\mathrm{m\,s^{-2}} \)

Speed of light\( c=3.00\times10^8\,\mathrm{m\,s^{-1}} \)

Elementary charge\( e=1.60\times10^{-19}\,\mathrm{C} \)

Electron mass\( m_e=9.11\times10^{-31}\,\mathrm{kg} \)

Planck constant\( h=6.63\times10^{-34}\,\mathrm{J\,s} \)

Permittivity of free space\( \epsilon_0=8.85\times10^{-12}\,\mathrm{F\,m^{-1}} \)

Magnetic constant\( \mu_0=1.26\times10^{-6}\,\mathrm{N\,A^{-2}} \)

Boltzmann constant\( k_B=1.38\times10^{-23}\,\mathrm{J\,K^{-1}} \)

Section A: Electromagnetism

1. Hall probe on a rotating field mapper

15 marks

A Hall probe is fixed to the end of a rotating field-mapper arm. At one instant the sensor strip has its length along \(+\hat{\imath}\), its width \(w=1.20\,\mathrm{mm}\) along \(+\hat{\jmath}\), and its thickness \(t=0.200\,\mathrm{mm}\) along \(+\hat{k}\). A steady conventional current \(I=18.0\,\mathrm{mA}\) flows along \(+\hat{\imath}\). The local magnetic field to be mapped is \(\vec B=0.420\,\mathrm{T}\,\hat{k}\). The mobile carriers in the semiconductor are electrons with density \(n=3.20\times10^{22}\,\mathrm{m^{-3}}\).

(a)

4 marks

Find the electron drift velocity in vector form.

(b)

4 marks

Calculate the Hall electric field and state which side of the strip becomes negative.

(c)

3 marks

Derive the Hall voltage magnitude and calculate its value.

(d)

2 marks

A calibration run gives \(|V_H|=7.50\,\mathrm{mV}\). Infer the carrier density from this reading.

(e)

2 marks

Estimate the magnetic field produced by the probe current \(25.0\,\mathrm{mm}\) from the current path, and compare it with the field being mapped.

2. Charging and dumping a pulsed electromagnet

15 marks

A laboratory pulsed electromagnet is modelled as a coil with inductance \(L=0.240\,\mathrm{H}\) and resistance \(R=3.00\,\Omega\). At \(t=0\), a \(24.0\,\mathrm{V}\) dc supply is connected in series with the coil. Later, the supply is disconnected and the coil is connected across a \(12.0\,\Omega\) dump resistor, so the discharge resistance is \(15.0\,\Omega\) including the coil.

(a)

3 marks

Write the current-growth equation, then calculate the time constant and final current.

(b)

3 marks

Find the current \(0.120\,\mathrm{s}\) after the supply is connected.

(c)

4 marks

At the same instant, calculate \(dI/dt\) and the self-induced emf of the inductor, including its sign relative to the current rise.

(d)

2 marks

Find the magnetic energy stored in the coil at \(t=0.120\,\mathrm{s}\).

(e)

3 marks

If the dump circuit is connected when the current is \(6.21\,\mathrm{A}\), find the time for the current to fall to \(0.500\,\mathrm{A}\).

Section B: Relativity and Quantum Mechanics

3. Unstable particles between beamline counters

15 marks

A beam of unstable particles travels in the \(+x\) direction at speed \(0.960c\) between two counters. Counter A is at the start of a straight evacuated section and counter B is \(18.0\,\mathrm{m}\) downstream. The particles have proper mean lifetime \(\tau_0=32.0\,\mathrm{ns}\). In the rest frame of one particle, a decay product can be emitted along the beam direction with speed \(0.600c\).

(a)

3 marks

Calculate the Lorentz factor and the mean lifetime measured in the laboratory.

(b)

3 marks

Find the laboratory flight time between the counters and compare it with the dilated lifetime.

(c)

4 marks

Calculate the fraction of particles that reach counter B and the fraction that decay between the counters.

(d)

4 marks

Use relativistic velocity addition to find the laboratory velocities of products emitted forward and backward along the beam in the particle rest frame. State the frames used. The velocity-addition equation is \[ u_x=\frac{u_x'+v}{1+u_x'v/c^2}, \] where \(S'\) is the particle rest frame and \(S\) is the laboratory frame.

(e)

1 mark

Explain why the counter spacing would be badly underestimated if the proper lifetime were used directly in the laboratory.

4. Electron states on a quantum ring

15 marks

An electron is confined to a very thin conducting quantum ring of radius \(R=0.850\,\mathrm{nm}\). Its position around the ring is described by the angle \(\phi\), and the potential energy is constant around the ring. The spatial wavefunction must be single-valued after one complete circuit.

(a)

3 marks

State the boundary condition for the ring and explain its physical meaning.

(b)

4 marks

Use the periodic boundary condition to find the allowed values of \(m\) and normalize the states. The trial angular states are \[ \psi_m(\phi)=Ae^{im\phi}. \]

(c)

3 marks

Find the angular momentum eigenvalue and derive the energy levels. The angular-momentum operator is \[ \hat L_z=-i\hbar\frac{d}{d\phi} \] and the rotational kinetic energy is \[ E=\frac{L_z^2}{2I}, \] where \(I=m_eR^2\).

(d)

2 marks

Discuss the degeneracy of the \(+m\) and \(-m\) states and identify any exception.

(e)

3 marks

Calculate \(E_1\) and the transition energy for \(m=2\to m=1\), in joules and electronvolts.