PHAS0038: Electromagnetic Theory

PST Example Problems 2

These problems will be demonstrated during the Problem Solving sessions in Week 5. This document (without solutions) will be handed out as hardcopy during the PS sessions. Solutions will be typeset and added to and online version of this document after those sessions have taken place.

1. To be demonstrated in class: Consider a region in which the magnetic field B varies in space as a function of position r, but does not vary with time t. Now consider a closed curve Γ, which moves through this region, keeping its fixed shape. Each point on the curve thus moves with the same velocity vo. Show that the rate of change of magnetic flux through any surface bounded by Γ is given by the line integral 

2. To be attempted in class: Generalise the result from the previous question to consider a rectangular loop initially at rest in the xy plane, with its centre at the origin. The loop has edges of length b parallel to the x axis and length a parallel to the y axis. The loop is mounted on an axis of rotation which runs parallel to the y axis and passes through its centre. The loop is embedded in a uniform. magnetic field B = B0zˆ which is parallel to the z axis.

At time zero, the loop is set into rotational motion with angular velocity !0 about the y axis. What is the magnitude of the EMF induced around the loop when motion com-mences? The subsequent angular velocity of the loop is described by a function of time  (for an appropriately defined angle φ). If the loop has a fixed resistance R, can you derive an expression for the torque exerted on the loop at time t?

3. Consider a cylindrical bar magnet (ferromagnet), which has a large permanent magnetiza-tion M in its interior (M is parallel to the cylinder axis of symmetry). M is approximately uniform. and large such that, in the interior of the magnet, B ≈ µ0M.

The magnet starts to move along the x axis (i.e. its cylindrical axis of symmetry remains on the Cartesian x axis). It starts from a position with its centre at x = L (L > 0) and moves with uniform. velocity in the negative x direction. At time tC, it has passed halfway through a conducting loop which has a radius a just large enough to allow it through. It then passes through the loop and proceeds to the position x = −L. Assuming L >> a, draw a graph of how you would expect the magnetic flux through the loop to change with time, due only to the passage of the magnet, from t = 0 to t = 2tC. Draw a corresponding graph of the magnitude of the EMF induced around the loop as a function of time.

4. A single circular conducting loop of radius d<a is placed inside a long solenoid of cylindrical radius a. The plane of the loop is initially perpendicular to the z (symmetry) axis of the solenoid. The solenoid carries n turns of wire per unit length through which a steady current I is flowing. It is long enough that the ‘infinite length’ approximation for its interior magnetic field is valid.

The loop then starts rotating about one of its diameters, so that the smallest angle between the z axis and the plane of the loop is ξ = π/2 − Ωt, where Ω is a constant, and t is time. Using this information, derive expressions for:

(i) The time-dependent magnitude of the mutual inductance between the solenoid and the loop (assuming that the solenoid is the only significant source of magnetic flux through the loop). The mutual inductance M can be defined by the equality Φ = MI, where Φ is the magnetic flux through the loop. (For revision / introduction to mutual inductance and self inductance, see the additional notes on Moodle related to ‘Notes and Solved Problems on Inductance’).

(ii) The time-dependent magnitude of the electromotive force (EMF) induced in the ro-tating loop.