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?

 

[kerimek]

Hello,

Thank you for reaching out for assistance. As a scientist, my suggestion would be to start by using the given equations and integrating them to find the magnetic scalar potential, \Omega. From there, you can use the relationship \textbf{E} = - \nabla \Omega to find the electric field at the specified points. It may also be helpful to consider the boundary conditions and use the continuity equation for the electric displacement field, \nabla \cdot \textbf{D} = \rho_f, where \rho_f is the free charge density. I hope this helps guide you in the right direction. Good luck with your calculations!