Question 5.57: A magnetic dipole m = −m0 zˆ is situated at the origin, in a...

From Eq. 5.88, B _{ tot }=B_{0} \hat{ z }-\frac{\mu_{0} m_{0}}{4 \pi r^{3}}(2 \cos \theta \hat{ r }+\sin \theta \hat{ \theta }) .

Therefore B \cdot \hat{ r }=B_{0}(\hat{ z } \cdot \hat{ r })-\frac{\mu_{0} m_{0}}{4 \pi r^{3}} 2 \cos \theta=\left(B_{0}-\frac{\mu_{0} m_{0}}{2 \pi r^{3}}\right) \cos \theta .

B _{ dip }( r )= \nabla \times A =\frac{\mu_{0} m}{4 \pi r^{3}}(2 \cos \theta \hat{ r }+\sin \theta \hat{ \theta })                  (5.88)

This is zero, for all θ, when r = R, given by B_{0}=\frac{\mu_{0} m_{0}}{2 \pi R^{3}} , or 

R=\left(\frac{\mu_{0} m_{0}}{2 \pi B_{0}}\right)^{1 / 3} . Evidently no field lines cross this sphere.