PHAS0038: Electromagnetic Theory

Problem Sheet 1

Complete the following three questions - show and justify your full working. Submit your work by the end of Tuesday, Nov 1, 2022. You should preferably submit a file containing a digital scan of your legible written work, or a word-processed document, using the upload link on the Moodle page. (Please submit your work in a common format such as .doc, .docx, .pdf or .jpeg)

This problem sheet is for private use only, and should not be re-distributed.

Question 1

Consider an infinitely large pair of thin conducting plates which both lie parallel to the xy plane in a Cartesian coordinate frame. The two plates are separated vertically by a distance d, measured parallel to the z axis (see diagram below). The upper and lower plates carry uniform. surface densities of free charge of −σ and σ, respectively (with σ > 0). The space outside and between the plates is vacuum.

Figure 1: Part of system with two infinitely large, parallel, charged plates.

(a) Show that the electric displacement between the plates is given by D = σzˆ (where zˆ is a unit vector in the positive z direction). (You may assume zero electric field in the space which is not between the plates).         [4 marks]

(b) The space between the plates is now filled with a LIH (linear, isotropic, homogeneous) dielectric material of relative permittivity ϵr (where ϵr > 1). The free charge distribution on the plates does not change. Show that the surface density of polarization (bound) charge which now accumulates on the upper and lower faces of the dielectric is given by ±σ (ϵr − 1)/ϵr. As part of your answer, indicate which algebraic sign in this expression corresponds with the upper plate and lower plate.             [6]

(Question 1 total marks: 10)

Question 2

A particle detector onboard a spacecraft can be represented as a rectangular box whose eight ver-tices are situated at Cartesian coordinates (x, y, z) = (0, 0, −D),(L, 0, −D),(0, W, −D),(L, W, −D), (0, 0, D),(L, 0, D),(0, W, D),(L, W, D) where L, W and 2D are the edge lengths of the box along the x, y and z directions. The electric field inside the detector is uniform, and given by E0 zˆ where E0 is a positive constant.

Ions from the space environment can enter a small aperture on one side of the container, which is situated at coordinates (0, W/2, 0).

Consider an ion of positive charge q and mass m which enters the aperture with initial velocity u0 ˆx at time t = 0. The only force acting on the ion after this time is due to the electric field inside the detector.

(a) Using the information above, show that the equations of motion of the ion (which has general velocity u(t)ˆx + v(t)zˆ, a vector function of time) are:

u ′ (t) = 0,

v ′ (t) = q E0/m.                          [2]

(b) The motion of this ion stops inside the detector when it collides with one of the walls at the point (x, y, z) = (xf , W/2, D) where 0 < xf < L. Show that the ion’s initial kinetic energy, at time t = 0, is given by [10]

(Question 2 total marks: 12)

Question 3

The planet Jupiter is surrounded by a thin, partial disc of plasma which carries electric current. In a cylindrical coordinate system with the centre of Jupiter at the origin (see diagram below), the current disc lies in the plane z = 0, and extends between cylindrical radial distances a (inner edge) and b (outer edge).

Figure 2: View from above Jupiter’s magnetic equator. Current disc extends between radii a and b, and carries current which is locally azimuthally directed.

The surface current density at any point in the disc is given by where I0 is a positive constant, R is cylindrical radial distance, and ˆϕ is a unit vector in the local azimuthal direction. jS has units of azimuthal current per unit radial length, so that the azimuthal current flowing across a small increment δR, at radial distance R, is (I0/R) δR.

(a) Consider an observer on the positive z axis, situated at a distance z0 from the origin (Jupiter’s centre). Use the Biot-Savart Law to show that the contribution to the z component of the magnetic field at the observer’s position, due to a small surface element R δR δϕ of the disc, is given by:

[9]

(b) Explain why the total magnetic field at the observer’s position, due to all of the disc, is directed along the z axis.          [2]

(c) Using the results from (a) and (b), show that the total magnetic field at the observer’s position is:

[7]

(Question 3 total marks: 18)