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. Guarded parallel-plate sensor

15 marks

A guarded parallel-plate sensor has plate area \(A=4.00\times10^{-4}\,\mathrm{m^2}\) and separation \(d=1.50\,\mathrm{mm}\). The guard ring makes edge effects negligible, so the field between the active plates may be treated as uniform. The sensor is first connected to a \(120\,\mathrm{V}\) supply in air.

(a)

3 marks

Find the electric field magnitude between the plates and state its direction.

(b)

4 marks

Calculate the capacitance in air and the magnitude of the charge on either active plate.

(c)

3 marks

Find the stored energy in the air-filled sensor.

(d)

5 marks

A dielectric slab with relative permittivity \(\kappa=3.20\) is inserted so that it completely fills the gap. Compare the new charge, field, voltage, and stored energy for the cases where the sensor remains connected to the supply and where it is first isolated.

2. Motional emf and magnetic braking

15 marks

A conducting rod of length \(\ell=0.250\,\mathrm{m}\) slides without friction on two horizontal rails. The rails are joined at the left by a resistor of resistance \(R=0.800\,\Omega\). The rod moves to the right at constant speed \(v=4.00\,\mathrm{m\,s^{-1}}\). A uniform magnetic field of magnitude \(B=0.600\,\mathrm{T}\) is directed into the page in the region occupied by the rod; outside that region the field is negligible.

(a)

3 marks

Find the motional emf across the rod and identify which end of the rod is at higher potential while the rod is in the field.

(b)

4 marks

Calculate the induced current and use Lenz's law to state its direction as the rod enters the field region.

(c)

3 marks

Find the magnetic braking force on the rod, including its direction.

(d)

3 marks

Compare the mechanical power needed to maintain the rod's motion with the electrical power dissipated in the resistor.

(e)

2 marks

State what happens to the emf and current after the rod has left the field region, and describe the Lenz-law direction during the exit interval.

Section B: Relativity and Quantum Mechanics

3. Synchronized detector gates on a shuttle

15 marks

A shuttle moves in the \(+x\) direction at \(v=0.600c\) relative to a station frame \(S\). In the shuttle frame \(S'\), two synchronized detector gates are separated by \(L'=600\,\mathrm{m}\). The rear gate is at \(x'=0\) and the front gate is at \(x'=600\,\mathrm{m}\). Both gates fire at \(t'=0\). The origins coincide at the rear-gate firing event. The inverse Lorentz transformations from the shuttle frame to the station frame are \[ x=\gamma(x'+vt') \] and \[ t=\gamma\left(t'+\frac{vx'}{c^2}\right). \]

(a)

5 marks

Transform the two gate-firing events into the station frame \(S\).

(b)

3 marks

State the time order of the two gate-firing events in each frame and explain the difference.

(c)

4 marks

At the instant the rear gate fires, it sends a light signal toward the front gate. Find the signal arrival event in both frames.

(d)

3 marks

Find the proper time elapsed on the front-gate clock between its firing and the arrival of the rear-gate signal, and check it against the station-frame interval.

4. Finite measurement window wavefunction

15 marks

A particle is known to be inside a finite measurement window \(0\le x\le L\). A trial spatial wavefunction is proposed as \(\psi(x)=Ax(L-x)\) inside the window and \(\psi(x)=0\) outside it.

(a)

3 marks

State the wavefunction outside the window and write the probability density inside the window. Check the boundary values.

(b)

4 marks

Normalize the trial wavefunction and find \(A\) in terms of \(L\).

(c)

4 marks

Find the probability of detecting the particle in \(0\le x\le L/2\) and in \(L/4\le x\le3L/4\).

(d)

4 marks

Estimate the position uncertainty from the normalized distribution. For \(L=0.500\,\mathrm{nm}\), estimate the lower bound on \(\Delta p\) and the corresponding kinetic-energy scale in electronvolts. The position uncertainty is \[ \Delta x=\sqrt{\langle x^2\rangle-\langle x\rangle^2} \] and the Heisenberg uncertainty relation is \[ \Delta x\Delta p\ge\frac{\hbar}{2}. \]