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

Exercise 4.1.1

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 4.1.1

Lorentz force in components

cf. the Lorentz force

A charge $q = 2$ μC moves with the velocity $v = (3 a_{x} + 4 a_{z}) \times 1 0^{5}$ m/s through a region where $E = 2 a_{y}$ kV/m and $B = 0.5 a_{z}$ T. Find (a) $v \times B$ , (b) the electric, the magnetic and the total force, (c) the acceleration if $m = 1 0^{- 12}$ kg, (d) the power delivered to the charge by each field.

Try it yourself first.

Exercise 4.1.1: Lorentz force in components