# Electric Field from Permanent Magnet

• insomniac392
In summary, the conversation is about finding the electric field at certain points given a slab of magnetized matter and a spinning magnetized sphere. The equations involved are related to the magnetic scalar potential and the magnetic charge density. The solution involves using a duality transformation to solve for the electric field.
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.

Last edited:
insomniac392 said:
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?

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!

## 1. What is an electric field from a permanent magnet?

The electric field from a permanent magnet is the force field created by the movement of electrons within the magnet. This field can interact with other objects and particles, causing them to move or experience a force.

## 2. How is the electric field from a permanent magnet different from an electric field from a charged particle?

An electric field from a permanent magnet is different from an electric field from a charged particle because it is created by the movement of electrons within the magnet, rather than a stationary charge. Permanent magnets have a north and south pole, and the electric field from a permanent magnet is always perpendicular to the direction of the magnetic field.

## 3. What factors affect the strength of the electric field from a permanent magnet?

The strength of the electric field from a permanent magnet is affected by the strength of the magnet, the distance from the magnet, and the material the magnet is interacting with. Additionally, the orientation of the magnet, as well as the presence of other magnetic fields, can also affect the strength of the electric field.

## 4. How can the electric field from a permanent magnet be measured?

The electric field from a permanent magnet can be measured using a device called a Hall probe, which measures the magnetic field and calculates the electric field based on the strength and orientation of the magnet. Other methods include using a compass or a magnetometer.

## 5. What are some applications of the electric field from a permanent magnet?

The electric field from a permanent magnet has many practical applications, including in electric motors, generators, and speakers. It is also used in magnetic levitation technology and particle accelerators. Additionally, the electric field from a permanent magnet can be used in medical imaging, such as MRI machines, to create detailed images of the body.

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