Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Exercise 4.2.9

About the further exercises of Section 4.2

Ten further exercises in the spirit of Problems 4.2.1–4.2.5, the worked examples and the home exercises, graded from easy to harder. Exercises 4.2.1–4.2.3 are about magnetic media and the boundary conditions, Exercises 4.2.4–4.2.5 about Ohm's law and the Joule phenomenon, Exercises 4.2.6–4.2.8 about inductance and energy, and Exercises 4.2.9–4.2.10 about dipoles and coupled circuits. They use only

B = μ_{0} H + M = μ H, J_{m} = \nabla \times \frac{M}{μ _{0}}, J_{m s} = \frac{1}{μ _{0}} M \times a_{n}, H_{1 τ} - H_{2 τ} = J_{n S}, B_{1 n} = B_{2 n}, L_{12} = \frac{N _{2} Φ _{12}}{I _{1}}, W_{m} = \frac{1}{2} \int H \cdot B d V = \frac{1}{2} L I^{2}, J = σ E, p = J \cdot E, T = m \times B .

As in the lectures, $M$ carries the units of $B$ (tesla), so $M / μ_{0}$ is the magnetisation in A/m.

Exercise 4.2.9

Compass needle in the Earth's field

cf. magnetic dipole and torque

A compass needle has the magnetic moment $m = 0.05 A \cdot m^{2}$ and the moment of inertia $J = 2 \times 1 0^{- 8} kg \cdot m^{2}$ about its pivot. The horizontal component of the Earth's field is $B = 50 μ T$ . (a) Find the torque when the needle makes $3 0^{\circ}$ with $B$ . (b) Find the potential energy $U = - m \cdot B$ and the work needed to turn the needle from the aligned to the reversed position. (c) Show that small oscillations about the aligned position are harmonic and find their frequency.

Try it yourself first.

Exercise 4.2.9: Compass needle in the Earth's field