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?