Permeable sphere placed in external magnetic field

[unscientific]

Homework Statement

A permeable sphere with $$μ_r$$ is placed in a magnetic field of $$H$$. Show that it satisfies Laplace's equation and the potentials are of the form:

Homework Equations

The Attempt at a Solution

Part (a)

Starting from maxwell's equations, I showed that ø satisfies laplace's equation. Followed by relating it to the electric case, where the general solution is similar.

Part(b)
I've only managed to get 2 useful boundary conditions..

Am I allowed to say that as $$r→ ∞, ø_2 → -H_0r cos θ$$ (Just as the electric case)

Because this would immediately solve for $$C_2 = H_0$$

Part (c)
To find dipole moment m, we use the the formula for dipole scalar potential for r > r₀:

$$ø_2 = -C_2 r cos θ + C_3 r^{-2} cos θ = \frac {\vec {m} .\vec {r}}{4\pi r^3}$$

 

[TSny]

Your equation (2) has a small error. Check the power of the $a$ in the denominator of the right side.

Am I allowed to say that as $$r→ ∞, ø_2 → -H_0r cos θ$$ (Just as the electric case)

Because this would immediately solve for $$C_2 = H_0$$

Yes, I believe that's right.

 

[unscientific]

Your equation (2) has a small error. Check the power of the $a$ in the denominator of the right side.



Yes, I believe that's right.

For part (c),

$$Magnetization = NM_0$$

How do we find the effective permeability?

I know that:

$$B = μM$$ where M is magnetization.

Do I find an expression for the field of a dipole very far away?

 

[TSny]

If you have obtained a result for the magnetic dipole moment m₀ of the paramagnetic sphere, then you should see that the moment is proportional to the applied field H.

Then when you look at the composite material with N of these spheres per unit volume, the total magnetization of the material will be M = Nm₀, as you say. Since m₀ is proportional to H, so is M.

Then consider how the proportionality constant between M and H is related to the effective permeability μ_eff.

This might help: http://en.wikipedia.org/wiki/Magnetic_susceptibility.

[There are some subtleties associated with the fact that in the composite material, the H-field felt by an individual paramagnetic sphere is not generally equal to the (averaged) macroscopic H field in the material. In the electric case, this leads to the Clausius-Mossotti equation. But in your problem, you are assuming that the spheres are "dilute" and so you can probably neglect this effect.]

 

[unscientific]

If you have obtained a result for the magnetic dipole moment m₀ of the paramagnetic sphere, then you should see that the moment is proportional to the applied field H.

Then when you look at the composite material with N of these spheres per unit volume, the total magnetization of the material will be M = Nm₀, as you say. Since m₀ is proportional to H, so is M.

Then consider how the proportionality constant between M and H is related to the effective permeability μ_eff.

This might help: http://en.wikipedia.org/wiki/Magnetic_susceptibility.

[There are some subtleties associated with the fact that in the composite material, the H-field felt by an individual paramagnetic sphere is not generally equal to the (averaged) macroscopic H field in the material. In the electric case, this leads to the Clausius-Mossotti equation. But in your problem, you are assuming that the spheres are "dilute" and so you can probably neglect this effect.]

$$M = xH = Nm_0 = gH$$ where g is the proportionality constant I have found earlier

$$μ_{eff} = μ_0(1+x)$$

Then solve for x in terms of g. Then plug it inside $$μ_{eff}$$

 

[TSny]

$$M = xH = Nm_0 = gH$$ where g is the proportionality constant I have found earlier

Maybe a factor of N is missing in the last expression on the far right?

$$μ_{eff} = μ_0(1+x)$$

Then solve for x in terms of g. Then plug it inside $$μ_{eff}$$

I think that's right.