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

Exercise 4.1.12

About the further exercises of Section 4.1

Twenty further exercises in the spirit of the problems, the worked examples and the home exercises, graded from easy to harder within each group. Exercises 4.1.1–4.1.10 belong to the theory of Lecture 8 (the Lorentz force, Biot–Savart's law, flux, force and the vector potential) and Exercises 4.1.11–4.1.20 to the exercise session of Lecture 9 (Ampère's law, coils and rotating charges, measuring and shaping fields). They use only

F = q (E + v \times B), F = I \int d l \times B, B = \frac{μ _{0}}{4 π} \int \frac{I d l ^{'} \times R}{R ^{3}}, B = \nabla \times A, \nabla \cdot B = 0, \oint_{C} H \cdot d l = I_{enc}, Φ = \int_{S} B \cdot d S,

the two results of the section, $B = μ_{0} I / (2 π ρ)$ for a long straight wire and $B_{z} = μ_{0} I b^{2} / [2 (b^{2} + h^{2})^{3/2}]$ on the axis of a loop, and superposition.

Exercise 4.1.12

Field in and around a copper wire

cf. Problem 4.1.5

A copper wire of radius $b = 2$ mm carries $I = 50$ A, uniformly distributed. Find (a) $H$ and $B$ at the surface, (b) $B$ at $ρ = 1$ mm inside the wire and at $ρ = 1$ cm outside, and (c) the radius at which $B$ has dropped to the Earth's field, $50$ μT.

Try it yourself first.

Exercise 4.1.12: Field in and around a copper wire