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

Exercise 4.2.2

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 4.2.2

Filling a solenoid with a material

cf. susceptibility and the relative permeability

A long solenoid has $n = 2000$ turns per metre and carries $I = 2 A$ . Find $H$ , $B$ and $M$ inside it when the core is (a) air, (b) a paramagnetic material with $χ_{m} = 2 \times 1 0^{- 3}$ , (c) soft iron with $μ_{r} = 200$ (well below saturation). What does the material change, and what does it not change?

Try it yourself first.

Exercise 4.2.2: Filling a solenoid with a material