Chapter 8.4, Problem 68E

Field Strength

The field strength H of a magnet of length 2L on a particle r units from the centre of the magnet is

$H=\frac{2mL}{{\left(r^2+L^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

Where $\pm m$ are the poles of the magnet. Find the average field strength as the particle moves from 0 to R units from the centre by evaluating the integral.

$\frac{1}{R}{\displaystyle \underset{0}{\overset{R}{\int }}\frac{2mL}{{\left(r^2+L^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}dr.$