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

Exercise 4.2.5

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 4.2.5

Radial current in a hollow cylinder

cf. Ohm's law

A hollow cylinder of inner radius $a = 1 cm$ , outer radius $b = 5 cm$ and length $l = 10 cm$ is made of a poorly conducting material, $σ = 10 S/m$ . Its inner and outer surfaces are coated with metal, and a current $I = 1 A$ flows radially from the inner to the outer surface. Find (a) $J$ and $E$ inside the material, (b) the voltage between the electrodes and the resistance, (c) the dissipated power, from $\int J \cdot E d V$ and from $I^{2} R$ .

Try it yourself first.

Exercise 4.2.5: Radial current in a hollow cylinder