# Meissner Effect

1. Oct 2, 2007

### big man

1. The problem statement, all variables and given/known data
Consider a sphere of type 1 superconductor with critical field Hc. (a) Show that in the Meissner regime the effective magnetisation M ithin the sphere is given by $$-3B_a/2\mu_0$$ (where Ba is the uniform applied magnetic field).

(b) Show that the magnetic field at the surface of the sphere in the equatorial plane is $$3B_a/2$$. (It follows that the applied field at which the Meissner effect starts to break down is $$2H_c/3$$.

Reminder: The demagnetisation field of a uniformly magnetised sphere is -M/3.

2. Relevant equations

$$H = H_a - H_d$$

$$H_a = B_a/ \mu_0 + M$$

$$m = MV$$ magnetic field of a magnetic dipole where V is the volume of the sphere

3. The attempt at a solution

The first part of this question is quite easy. I'm given the demagnetisation field for a uniformly magnetised sphere and I know that with the Meissner effect in superconductors, the field inside must be equal to 0. So I used the expression for the applied field (second equation) and substituted it into the first equation with Hd = -M/3. This was then set to 0 and I solved for M.

The second part of the question is what I really need help on. I don't really understand how to do this at all. I know that the magnetic field outside of the sphere is the magnetic field of a magnetic dipole, but I don't really know the significance of this. It was just a hint that we were meant to use this.

i.e. $$m = (4 \pi/3 )r^3M$$ r is the radius of the sphere

I'd appreciate any ideas on how to approach this second part of the problem. I'm sorry that I don't have much of an attempt for this, but I just don't have an idea of how to start this second part. You obviously need to consider that you are in equatorial plane, but I just can't see how to do this.

Last edited: Oct 2, 2007