Magnetic field of a rotating disk with a non-uniform volume charge

[Light bulB 6626]

Homework Statement

Consider a thin disk ( $d \ll R$ , with thickness $d$ and radius $R$ ), with volume

charge distributed as $\rho_{f}=k(R-s)(d-z)^{2}$ , where $s=\sqrt{x^{2}+y^{2}}$ and the coordinate origin $(x, y, z)=(0,0,0)$ is at the center of the disk. (The $z$ axis is normal to the circular bases of disk). When this disk rotates around
the $z$ axis with angular speed $\omega,$ answer the following questions:

(a) Find the free current density $\vec{J}_{f}$

(b) Show that $\nabla^{2} \vec{B}=-\mu_{0} \nabla \times \vec{J}_{f}$ is satisfied in this case.

(c) Using (b), find $\vec{B}$ near the disk, at $z \simeq d / 2 \ll R$

Relevant Equations

$\nabla^{2} \vec{B}=0$ (outside the disk).
$\nabla^{2} \vec{B}=-\mu_{0} \nabla \times \vec{J}_{f}$ (inside the disk)

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This was a problem introduced during my classical electrodynamics course.
I am not 100% sure, but I think I've solved up to problems (a) and (b) as below.
(Note that I tried to follow the notations in Introduction to Electrodynamics (Griffiths) :(a) We use cylindrical coordinates $(s, \phi, z)$ , with unit vectors $\hat{r}, \hat{\phi}, \hat{z}$ .
$\vec{J}_{f} = \vec{J} = \rho \vec{v} = \rho s w \hat{\phi}$ .

$\therefore \vec{J}_{f}=k(R-s)(d-z)^{2} \rho s w \hat{\phi}$

(*I am not absolutely sure whether $\vec{J}_{f} = \vec{J}=\vec{J}_{f}+\vec{J}_{b}+\vec{J}_{p}$ holds in this problem.)

(b) $\nabla \times \vec{B} = \mu_{0} \vec{J}$ (Ampere's law)

$\nabla \cdot \vec{B} = 0$
( $\because$ there are no magnetic monopoles)

$\nabla \times (\nabla \times \vec{B}) = - \nabla^{2} \vec{B} + \nabla (\nabla \cdot \vec{B})=- \nabla^{2} \vec{B}$

$\therefore \nabla^{2} \vec{B}=-\mu_{0} \nabla \times \vec{J}_{f}$*****************************************************************

> I have been pondering on question (c) for more than a full day, but I couldn't get a clear solution. Here are some of my (partial) attempts:

1. **Laplace equation for cylindrical coordinates**.

$\nabla^{2} \vec{B}=0$ (outside the disk).Solve the Laplace equation for components of $B$ , with boundary conditions obtained from

$\nabla^{2} \vec{B}=-\mu_{0} \nabla \times \vec{J}_{f}$ (inside the disk)

> I haven't tried this approach since because I am not familiar with solving Laplace equations for cylindrical coordinates, (also, the calculations seemed too tedious and complicated for the problem.)

2. Greens function method.

(I am not familiar with this method, but a friend of mine (a graduate from my school) advised me this method. I do not perfectly understand the Greens function method, but I was able to come up with this formula, using **some analogies** (with $V$ and $\vec{A}$ ) from Introduction to Electrodynamics* (Griffiths).$\nabla^{2} V=-\frac{\rho}{\epsilon_{0}}$ $\Longrightarrow$
$V(\mathbf{r})=\frac{1}{4 \pi \epsilon_{0}} \int \frac{\rho\left(\mathbf{r}^{\prime}\right)}{|\vec{r} - \vec{r}^{\prime}|} d \tau^{\prime}$

$\nabla^{2} \mathbf{A}=-\mu_{0}\mathbf{J}$
$\Longrightarrow$ $\mathbf{A}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right)}{|\vec{r} - \vec{r}^{\prime}|} d \tau^{\prime}$

Therefore
$\nabla^{2} \mathbf{B}=-\mu_{0} \nabla \times \mathbf{J}_{f}$
$\Longrightarrow$
$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int \frac{(\nabla \times \mathbf{J}_{f})\left(\mathbf{r}^{\prime}\right)}{|\vec{r} - \vec{r}^{\prime}|} d \tau^{\prime}$

I have tried this method, but was unable to proceed since the integral (if my calculations were correct) involved an elliptic integral. I tried to use the approximation $z \simeq d / 2 \ll R$ to simplify the integral (- expanding it using the taylor expansion, then keeping terms upto order 1), but couldn't find a subtle arrangement of terms.

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I am a undergraduate, and I am still learning/practicing how to approach these kind of problems. Any suggestions about solving the problem welcome. I would be very grateful if some kind user will provide a solution/ or a detailed advice for problem (c). Thank you.

 

[Paul Colby]

You might look at Stokes Theorem. The $\nabla^2\mathbf{B}$ clearly has to point along $z$ so look at the area integral in the xy plane. Not sure this will get you much but the lhs of (b) was made for this kind of thing.

P.S. you can also use cylindrical symmetry. This screams make the area integrated a disk of radius $s$ for any value of $s$.

[edit] because $\rho(s,z)$, $\nabla^2\mathbf{B}$ will also have a $\hat{s}$ component.