Introduction to Electrodynamics · Problem 6.5

Magnetic Fields in Matter — David J. Griffiths

Physics · force-magnetic-dipole

Problem

A uniform current density

\mathbf{J} = J_0\hat{z}

fills a slab straddling the

yz

plane, from

x = -a

x = +a

. A magnetic dipole

\mathbf{m} = m_0\hat{x}

is situated at the origin. (a) Find the force on the dipole, using Eq. 6.3. (b) Do the same for a dipole pointing in the

y

direction:

\mathbf{m} = m_0\hat{y}

. (c) In the electrostatic case, the expressions

\mathbf{F} = \nabla(\mathbf{p}\cdot\mathbf{E})

and

\mathbf{F} = (\mathbf{p}\cdot\nabla)\mathbf{E}

are equivalent (prove it), but this is not the case for the magnetic analogs (explain why). As an example, calculate

(\mathbf{m}\cdot\nabla)\mathbf{B}

for the configurations in (a) and (b).