Exercise 4.2.7: Mutual inductance of two distant coaxial loops

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 4.2.7

Mutual inductance of two distant coaxial loops

cf. the magnetic dipole

Two circular loops are coaxial: loop 1 has radius $a_{1} = 2 cm$ , loop 2 has radius $a_{2} = 1 cm$ , and their centres are $d = 20 cm$ apart. (a) Using the dipole field of Section 4.2.8, find $B_{1}$ on the axis at the distance $d$ for a current $I_{1}$ . (b) Find $Φ_{12}$ and $L_{12}$ . (c) Which way would you compute $L_{21}$ , and why must it be the same?

Try it yourself first.