Classical Electrodynamics · Problem 5.4

Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws — John David Jackson

Physics — E&M · magnetic-induction-off-axis-expansion

Problem

A magnetic induction

\mathbf{B}

in a current-free region in a uniform medium is cylindrically symmetric with components

B_z(\rho, z)

and

B_\rho(\rho, z)

and with a known

B_z(0, z)

on the axis of symmetry. The magnitude of the axial field varies slowly in

z

. (a) Show that near the axis the axial and radial components of magnetic induction are approximately

B_z(\rho, z) \approx B_z(0, z) - \left(\frac{\rho^2}{4}\right)\left[\frac{\partial^2 B_z(0,z)}{\partial z^2}\right] + \cdots

B_\rho(\rho, z) \approx -\left(\frac{\rho}{2}\right)\left[\frac{\partial B_z(0,z)}{\partial z}\right] + \left(\frac{\rho^3}{16}\right)\left[\frac{\partial^3 B_z(0,z)}{\partial z^3}\right] + \cdots

(b) What are the magnitudes of the neglected terms, or equivalently what is the criterion defining ``near'' the axis?