Electric Field from Permanent Magnet

[insomniac392]

Hello,

I've been extremely stuck on the following problems and was hoping someone could give me a push in the right direction:

Homework Statement

1) Given an infinite slab of permanently magnetized matter of thickness d centered on the xy-plane with uniform magnetization \textbf{M} = (0, M, 0) and velocity \textbf{v} = (v, 0, 0), find the electric field at \textbf{E}(0, 0, 0) and \textbf{E}(0, y, 0) where y > d.

2) A magnetized sphere with uniform magnetization \textbf{M} = (0, 0, M) and radius r is spinning at a rate of \textbf{\omega} = (0, 0, \omega). Find the electric field for r' > r. (Hint: Find the equivalent magnetic charge density, \rho_m, and equivalent surface current, \sigma_m.)

Homework Equations

I'm not entirely sure (hence the thread)!

\sigma_{m, n} = \textbf{M} \cdot \textbf{n}

\rho_{m} = - \nabla \cdot \textbf{M}

...these are factors of the integrand that give rise to the magnetic scalar potential, \Omega, which in turn yields \textbf{B} via \textbf{H} = - \nabla \Omega.

The Attempt at a Solution

I'm desperately stuck on these; for both problems I can find \rho_m and \sigma_m, but I don't see the connection to the \textbf{E}-field. Any suggestions to get me started would be greatly appreciated.

 

[insomniac392]

Hello,

I've been extremely stuck on the following problems and was hoping someone could give me a push in the right direction:

Homework Statement

1) Given an infinite slab of permanently magnetized matter of thickness d centered on the xy-plane with uniform magnetization \textbf{M} = (0, M, 0) and velocity \textbf{v} = (v, 0, 0), find the electric field at \textbf{E}(0, 0, 0) and \textbf{E}(0, y, 0) where y > d.

2) A magnetized sphere with uniform magnetization \textbf{M} = (0, 0, M) and radius r is spinning at a rate of \textbf{\omega} = (0, 0, \omega). Find the electric field for r' > r. (Hint: Find the equivalent magnetic charge density, \rho_m, and equivalent surface current, \sigma_m.)

Homework Equations

I'm not entirely sure (hence the thread)!

\sigma_{m, n} = \textbf{M} \cdot \textbf{n}

\rho_{m} = - \nabla \cdot \textbf{M}

...these are factors of the integrand that give rise to the magnetic scalar potential, \Omega, which in turn yields \textbf{B} via \textbf{H} = - \nabla \Omega.

The Attempt at a Solution

I'm desperately stuck on these; for both problems I can find \rho_m and \sigma_m, but I don't see the connection to the \textbf{E}-field. Any suggestions to get me started would be greatly appreciated.

Well, after doing some digging I found the following problem (7.60) in Griffiths' Electrodynamics:

Maxwell's equations are invariant under the following duality transformations

\textbf{E'} = \textbf{E} cos(\alpha) + c \textbf{B} sin(\alpha)

c \textbf{B'} = c \textbf{B} cos(\alpha) - \textbf{E} sin(\alpha)

c q_{e}' = c q_{e} cos(\alpha) + q_{m} sin(\alpha)

q_{m}' = q_{m} cos(\alpha) - c q_{e} sin(\alpha)

...where c = 1/\sqrt{\epsilon_0 \mu_0}, q_m is the magnetic charge and \alpha is an arbitrary rotation angle in "\textbf{E}-\textbf{B} space."

Griffiths' says that, "this means, in particular, that if you know the fields produced by a configuration of electric charge, you can immediately (using \alpha = \pi / 2) write down the fields produced by the corresponding arrangement of magnetic charge."

Thus, if I were to solve for \textbf{E} in (1) and (2) with a polarization \textbf{P} instead of a magnetization \textbf{M}, I could then use a duality transformation to find the solutions to (1) and (2), correct?