On-Axis Field of a Uniformly Magnetized Sphere

[Luke Tan]

Homework Statement

Find the magnetic field of a sphere along the axis of magnetization

Relevant Equations

$$\vec{B}=\frac{\mu_0}{4\pi}\int\frac{\vec{K}\times\hat{r}}{|\vec{r}|^2}da$$
$$\vec{K}=\vec{M}\times\hat{n}$$

Since I am only required to find the on-axis field, I tried directly integrating the biot savart to find the field, rather than integrating to find the vector potential before taking the curl.

However, on integration (by mathematica) it seems that the solution is an elliptic integral, very different from what griffiths has found in his book. Can someone find the issue in my working? Thanks!

 

[Charles Link]

I found one error on the left side. You got a little sloppy with your diagram. $x \neq R \cos{\theta}$. $\$ Edit: Scratch that. I need to study this more...$\\$ I think your last line should have $\sin^3{\theta}$, instead of $\sin^4{\theta}$.$\\$ Writing $\sin^3{\theta} \, d \theta=(1-cos^2{\theta}) \, d(\cos{\theta})=(1-u^2) \, du$,$\\$ I think you will find the $u^2/(denominator)$ can be integrated by parts. $\\$ Doing it slightly differently, I also got a $\sin^3{\theta}$ instead of $\sin^4{\theta}$. That may be your only error.