PHYS2114 – Electromagnetism Term 2, 2020
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
FINAL (TAKE HOME) EXAMINATION
PHYS2114 – Electromagnetism
Term 2, 2020
Question 1 (15 marks)
Consider an infinitely long dielectric cylinder of radius R with axis of symmetry along the z-axis, which carries a “frozen in” fixed polarisation P = P0x, where P0 is a constant.
(a) Find the surface and volume bound charge density for the cylinder. (2 marks)
(b) Find the electric field inside the cylinder. (Hint: consider the field produced by induced surface charges on a cylindrical conductor in a uniform electric field, problem 3 of tutorial 4). (2 marks)
(c) Find the electric potential V (s,φ) for s < R. (1 mark)
(d) Find the electric potential V (s,φ) for s ≥ R. (4 marks). The general solution of Laplace’s equation in cylindrical coordinates when there is no dependence on z is given by
1
V (s,φ) = a0 +b0 lns+X [sk (ak coskφ + bk sinkφ)+ s−k (ck coskφ + dk sinkφ)]
k=1
(e) Find the electric field E(s,φ), for s > R. (2 marks)
(f) Verify that E(s,φ) satisifies the required boundary conditions at the surface of the dielectric cylinder.(4 marks)
Question 2 (10 marks)
Note: You must show your work or explanation for all parts of the ques- tion.
A long wire of radius a carries a current I, and is surrounded by a long hollow cylinder made of a linear magnetic material with magnetic susceptibility χm . The inner radius of the cylinder is b and the outer radius c.
(a) Find the surface bound current density on the inner and outer surfaces of the cylinder. What is the direction of each surface current relative to I if the material is diamagnetic. (2 marks)
(b) Find the volume bound current density inside the cylinder. (2 marks)
(c) Find the magnetic field B(s), where s is the distance from the axis of the wire, for all s. (4 marks)
(d) How would the magnetic field for s > c be a↵ected if the cylinder were removed? (2 marks)
Question 3 (15 marks)
An alternating voltage source is connected in series with a 500 turn toroidal inductor and a spherical resistor . The toroidal inductor has a square profile with an inner radius of 10 cm and an outer radius of 12 cm . The resistor consists of two thin concentric spherical conducting shells with an inner radius of 2 cm an outer radius of 4 cm and the space between the space if filled with a material that has a resistivity of 100 ⌦ .cm .
(a) Derive an expression for the magnetic field of the inductor and hence the induc- tance . (4 marks)
(b) Derive an expression for the electric field inside the resistor and hence calculate the resistance . (4 marks)
(c) Use Kirchho↵’s laws to derive a relationship between the current and voltage when a voltage of V0 cos(!t) is applied to the circuit . (4 marks)
(d) Find the frequency at which the voltage drop across each component is half the total applied voltage . (3 marks)
Question 4 (10 marks)
Two plane waves are propagating in a medium of refractive index n1 . The first wave travels along the x-axis and the second travels in the x-y plane at an angle of 45 degrees inclination from the x axis. The first wave is linearly polarised along the y-direction, the second wave is right hand circularly polarised. Both waves have an amplitude of E0 .
(a) What is the wave-vector of each plane wave (magnitude and direction)? (2 marks)
(b) Write an expression for the combined electric field in Euler’s form. (3 marks)
(c) Determine an expression for the energy density at x = 0 and sketch the result. (5 marks)
2022-08-10