Fera Academy

Paper 2

Time2 hours

Marks60

SetSet 3

PaperLevel 1 - Physics Paper 2

Information

Section A: Electromagnetism
Section B: Relativity and Quantum Mechanics

Candidate name

Candidate number

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.
Use the constants printed in this paper where relevant.
Answer spaces are provided after each question part.

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. Charged insulating sheet field cage

[15 marks]

Two square insulating sheets are mounted parallel to one another to make a simple field cage. Each sheet has area \(A=0.120\,\mathrm{m^2}\), and the separation is \(d=3.00\,\mathrm{mm}\). The left sheet has charge \(+Q=8.50\,\mathrm{nC}\) spread uniformly over it, and the right sheet has charge \(-Q\) spread uniformly over it. Treat the sheets as very large compared with their separation, so edge effects may be ignored. The field magnitude from one very large uniformly charged sheet is \[ E=\frac{\sigma}{2\epsilon_0}. \] The field is directed away from a positive sheet and toward a negative sheet.

a) Find the surface charge density and the electric field magnitude inside and outside the sheet pair. State the field direction inside the cage.

[4 marks]

b) Calculate the potential difference between the sheets and identify which sheet is at higher potential.

[3 marks]

c) Find the capacitance per unit area and the total capacitance of the sheet pair. Check your result using \(V=Q/C\).

[4 marks]

d) Calculate the electrostatic energy stored in the field between the sheets.

[2 marks]

e) The sheet pair is isolated and the gap is filled with a dielectric of relative permittivity \(\kappa=2.40\). State the new field, voltage, and capacitance per unit area.

[2 marks]

2. Crossed-field ion selector

[15 marks]

A beam of singly charged positive ions enters a velocity selector travelling in the \(+\hat{\imath}\) direction. In the selector, \(\vec E=1.80\times10^4\,\hat{\jmath}\,\mathrm{N\,C^{-1}}\) and \(\vec B=0.300\,\hat k\,\mathrm{T}\). Ions that pass undeflected then enter an analyser containing only a uniform magnetic field of magnitude \(0.500\,\mathrm{T}\). Two isotopes have masses \(m_1=3.32\times10^{-26}\,\mathrm{kg}\) and \(m_2=3.49\times10^{-26}\,\mathrm{kg}\).

a) Show that the electric and magnetic forces oppose one another, and find the selected ion speed.

[4 marks]

b) State the deflection direction for ions that are slower and faster than the selected speed.

[2 marks]

c) Calculate the circular analyser radius for each isotope.

[4 marks]

d) Find the difference in analyser radii. If the detector is placed after a semicircle, estimate the separation between the two impact points and identify which isotope lands farther from the entry point.

[3 marks]

e) For an analyser field in the \(+\hat k\) direction, state the initial bending direction. What changes if the analyser field is reversed?

[2 marks]

Section B: Relativity and Quantum Mechanics

3. Spacelike platform flashes

[15 marks]

Two camera flashes occur along a straight train platform. In the platform frame \(S\), flash A occurs at \(x_A=0\), \(t_A=0\), and flash B occurs at \(x_B=600\,\mathrm{m}\), \(t_B=1.00\,\mu\mathrm{s}\). Another inertial frame \(S'\) moves along \(+x\) relative to the platform. For a frame \(S'\) moving at speed \(u\) along \(+x\) relative to \(S\), the Lorentz transformation for separations is \[ \Delta t'=\gamma\left(\Delta t-\frac{u\Delta x}{c^2}\right) \] and \[ \Delta x'=\gamma(\Delta x-u\Delta t). \]

a) Calculate \(s^2=c^2\Delta t^2-\Delta x^2\) for the two flashes and classify the separation.

[3 marks]

b) Find the speed and direction of the inertial frame in which the flashes are simultaneous.

[4 marks]

c) Calculate the transformed spatial separation of the flashes in the frame found in part (b).

[3 marks]

d) For a frame moving at \(0.800c\) in the \(+x\) direction, find \(\Delta t'\) and state the time order of the flashes.

[3 marks]

e) Explain why the change of time order in some frames does not violate causality.

[2 marks]

4. Infinite well spectroscopy after widening

[15 marks]

An electron is confined in a one-dimensional infinite square well. The original well width is \(L_0=0.500\,\mathrm{nm}\), but the well is then widened to \(L=0.750\,\mathrm{nm}\). For width \(L\), the stationary-state energies are \[ E_n=\frac{n^2h^2}{8m_eL^2} \] and the normalized wavefunctions are \(\psi_n(x)=\sqrt{2/L}\sin(n\pi x/L)\) for \(0<x<L\).

a) For the widened well, calculate \(E_1\) in joules and electronvolts, then find \(E_2\) and \(E_3\) in electronvolts.

[4 marks]

b) Find the wavelength of the photon emitted when an electron falls from \(n=3\) to \(n=2\) in the widened well.

[3 marks]

c) Write \(\psi_2(x)\) and calculate the probability of finding the electron between \(x=L/4\) and \(x=L/2\).

[4 marks]

d) By what factor do all energy levels change when the well is widened from \(0.500\,\mathrm{nm}\) to \(0.750\,\mathrm{nm}\)?

[2 marks]

e) State how the photon wavelength for the same transition changes after widening the well, and explain the spectral shift.

[2 marks]

END