Solving for M in a Type 1 Superconductor Sphere

[big man]

Homework Statement

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.

Homework 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

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.

 

[Jhero]

For the second part of the problem, we can use the fact that the magnetic field at the surface of a uniformly magnetised sphere is given by:

B = μ0(M+Hd)

Since we are in the equatorial plane, we can assume that the magnetic field at the surface of the sphere is only due to the demagnetisation field (Hd), since the applied field (Ha) is perpendicular to the surface.

So, we can rewrite the equation as:

B = μ0Hd = -μ0M/3

Now, we can substitute this into the equation for the demagnetisation field, Hd = -M/3, and solve for M:

M = -3B/2μ0

Finally, we can substitute this value of M into the equation for the magnetic field at the surface of the sphere, B = μ0(M+Hd), and solve for B:

B = μ0(-3B/2μ0 + -3B/2) = μ0(-3B/2μ0 + 3B/2) = 3B/2

Therefore, the magnetic field at the surface of the sphere in the equatorial plane is 3Ba/2, and the applied field at which the Meissner effect starts to break down is 2Hc/3.