How Does Material Permeability Affect Magnetic Field Shielding and Intensity?

[king vitamin]

Homework Statement

Consider a uniform B_{0} in the z direction, into which a hollow sphere is placed. If the sphere is made of a material with a high permeability (B = \mu \ H), the magnetic field inside will be greatly reduced. Calculate the magnetic field inside the sphere and the maximum field in the metal.

(my professor talks at length about the method of solving using laplace's eqn., etc)

You will find the shielding improves with \mu, but the maximum field in the metal also increases.

The Attempt at a Solution

I've already done all the grunt work - that is, I've calculated the field in all three regions (outside the sphere, inside the sphere, and in the spherical shell). The problem is actually similar to an example in Jackson (page 203), which gives expressions for the field inside and outside which agree with what I obtained, and conclude that the magnetic field inside the sphere decreases as 1/\mu.

The problem is the second part of the problem - that the maximum magnetic field inside the metal increases with \mu. The expression I obtain for the magnetic field inside the shell (with a being the inner radius, b the outer radius, and \mu_{r}=\frac{\mu}{\mu_{0}}) is as follows:

\begin{equation}<br /> \frac{3 \ \mu_{r} \ B_{0}}{(\mu_{r}+2)(2\mu_{r}+1) - 2\frac{a^{3}}{b^{3}}(\mu_{r}-1)^{2}} \ \{ ( \ (2 \mu_{r}+1)-\frac{2 a^{3} (\mu_{r}-1)}{r^{3}})cos \theta \mathbf{\hat{r}} - ( \ (2 \mu_{r} + 1) +\frac{a^3(\mu_{r}-1)}{r^{3}})sin \theta \mathbf{\hat{\theta}} \}<br /> \end{equation}<br />

To maximize this, I put in \theta = 0 and r = b, and the magnitude of the magnetic field (just the r component) is:

\begin{equation}<br /> \frac{3 \ \mu_{r} \ B_{0} \ \{(2 \mu{r} + 1) - 2 \frac{a^{3}}{b^{3}} (\mu_{r} - 1) \} }{(\mu_{r}+2)(2\mu_{r}+1) - 2\frac{a^{3}}{b^{3}}(\mu_{r}-1)^{2}}<br /> \end{equation}<br />

But as \mu gets large, this doesn't increase, it converges to 3B. Did I not find the maximum of the field in the metal correctly? Am I missing the question somehow? Maybe I took the gradient wrong? (I'm pretty certain I did the "potential" part of the problem correctly)

 

[kuruman]

I don't think it is appropriate to pick a value θ = 0 as you have. If I were doing this, I would find the magnitude of the B field first, then see what happens to it as μ gets large. Also, in your expression, are both unit vectors r-hat or should one of them be theta-hat?

 

[king vitamin]

Whoops, the second unit vector should be a theta-hat, I'll edit the post.

If the magnitude is just the r component of the vector, and the r component is proportional to the cosine of theta, then the maximum value would be where cosine is zero, right? Regardless, for some constant theta and r, the limit as mu increases still approaches a constant...

I'm starting to think the expression for the magnetic field is wrong, I'll post the scalar potential from which I derived this tomorrow.

 

[king vitamin]

I've started rethinking this; it doesn't make sense that a metal inside a uniform magnetic field would contain a magnetic field going to infinity inside the metal - I think the question was "show the magnetic field is amplified inside the metal," in which case my answer is correct - as mu gets large, the max field in the metal is 3 times larger than the initial field. I think I just overanalyzed it.