[Physics] Vector potential of a solenoid in the Coulomb gauge

electromagnetismelectrostaticsgaugepotential

I understand the usual argument for calculating the vector potential outside of a solenoid of radius $R$ with $n$ turns per unit length carrying current $I_0$ using
$$
\oint \mathbf{A} \cdot d \mathbf{l} = \iint \nabla \times \mathbf{A} \cdot d\mathbf{a} = \iint \mathbf{B} \cdot d\mathbf{a}
$$
which gives (in Gaussian units)
$$
A_{\varphi} = \frac{2\pi}{c} \frac{nI_0 R^2}{r}
$$
However, I am asked explicitly to find the vector potential in the Coulomb gauge. I have two main questions:

1) Is showing that this vector potential satisfies $\nabla \cdot \mathbf{A} = 0$ and $\mathbf{B} = \nabla \times \mathbf{A}$ sufficient? That seems a bit too much like a 'physicist proof' to me.

2) How can I compute the vector potential explicitly from the form
$$
\mathbf{A} = \frac{1}{c} \int \frac{\mathbf{J}(\mathbf{r}',t)}{|\mathbf{r}-\mathbf{r}'|} d^3 r
$$
I have written
$$
\mathbf{J}(\mathbf{r}',t) = n I_0 \frac{\delta(r'-R)}{R} \ \hat{\varphi}
$$
which gives after some algebra and one integration
$$
\mathbf{A} = \frac{n I_0}{c} \int_0^{2\pi} \int_{-\infty}^{\infty} \frac{1}{\sqrt{r^2+R^2 – 2rR \cos(\varphi-\varphi') + (z-z')^2 }}dz' d\varphi' \ \hat{\varphi}
$$
But doesn't the integral over $z$ diverge? This integral isn't doable by Mathematica. Have I done something wrong?

EDIT:

I suppose I can simplify this integral by (without loss of generality) letting $\phi = 0$ and $z=0$. The integral becomes
\begin{align*}
\mathbf{A} &= \frac{n I_0}{c} \int_0^{2\pi} \int_{-\infty}^{\infty} \frac{1}{\sqrt{r^2+R^2 – 2rR \cos(\varphi') + (z')^2 }}dz' d\varphi' \ \hat{\varphi} \\
&= \frac{n I_0}{c} \int_{-\infty}^{\infty} \frac{2 K\left(-\frac{4 r R}{(r-R)^2+(z')^2}\right)}{\sqrt{(r-R)^2+(z')^2}}+\frac{2 K\left(\frac{4 r R}{(r+R)^2+(z')^2}\right)}{\sqrt{(r+R)^2+(z')^2}} dz'
\end{align*}
But this still seems to diverge. How can I show that this reduces to $\frac{2\pi}{c} \frac{nI_0 R^2}{r} $?

Best Answer

  1. Yeah it is enough. Coloumb Gauge is defined by $\nabla \cdot \mathbf{A} = 0$. Also this expression for the vector potential was obtained through the solution of Poisson`s equation ($\nabla^2 \mathbf{A} = -\mu_0\mathbf{J}$), which already required $\nabla \cdot \mathbf{A} = 0$ (the full equation is $-\nabla(\nabla \cdot \mathbf{A})+\nabla^2 \mathbf{A} = -\mu_0\mathbf{J}$). So calculating through this expression already implies you will obtain an answer in the Coulomb Gauge.

First. In the integral at the end it's not "$\hat{\phi}$" that should be there, but "$\hat{\phi^{\prime}}$" since this is part of the current density and in the current density you put prime coordinates. And yeah, you integrate unitary vectors, as I will explain.

Second "$\hat{\phi^{\prime}}$" is not a constant of integration. Remenber that ALWAYS. Very easy to forget.

$\hat{\phi^{\prime}} = -\sin(\phi^{\prime})\hat{x} + \cos(\phi^{\prime})\hat{y}$

Unitary vectors are not constants in the integration when using curvilinear coordinates. Only cartesian unitary vectors are constants in the integration.

As you've said, put $z=0$ and $\phi=0$

Then you are good to go.

Calculation of the potential vector for the infinite cylindrical solenoid

Now I'm going to show how to calculate the vector potential of a Solenoid. Starting from:

$\mathbf{A} = \frac{\mu_0}{4\pi} \int \frac{\mathbf{K}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} d^2 r^\prime$

$d^2 r^\prime= Rdz^\prime d\phi^\prime$

Where the surface current density is:

$\mathbf{K}(\phi^\prime) =nI\hat{\phi^\prime} = nI( -sin(\phi^{\prime})\hat{x} + cos(\phi^{\prime})\hat{y})$

We will use the expansion:

$\frac{1}{|\mathbf{r}-\mathbf{r}'|}=$

$\frac{4}{\pi}\int_0^\infty dk\cos{[k(z-z^\prime)]}(\frac{1}{2}I_0(k\rho_<)K_0(k\rho_>)+\sum_{m=1}^\infty\cos{[m(\phi-\phi^\prime)]}I_m(k\rho_<)K_m(k\rho_>))$

Where $\rho_< = min[\rho,R]$ and $\rho_>=max[\rho,R]$. This expression can be obtained by finding an expansion for a Green function in cylindrical coordinates.

EDIT: For those coming from more elementary Electrodynamics, this expansion is a rather canonical procedure. The complexity of the computation of those kind of integrals lies almost always entirely in the 1/r² term, thus different expansions in infinite series for this term are used in order to perform the integration. The expansion is chosen based on the geometry of the problem. If your aren't expected to know that, than you're supposed to use some sort of approximation procedure in order to calculate the vector potential.

Now, using what I said previously is just a matter of simple integration. Using the fact that:

$\int_0^{2\pi}\sin(\phi^\prime)\cos{[m(\phi-\phi^\prime)]}d\phi^\prime=\pi\delta_{m,1}\sin(\phi)$

$\int_0^{2\pi}\cos(\phi^\prime)\cos{[m(\phi-\phi^\prime)]}d\phi^\prime=\pi\delta_{m,1}\cos(\phi)$

$\int_{-\infty}^{\infty}\cos{[k(z-z^\prime)]}dz^\prime=\Re{(e^{(ikz)}\int_{-\infty}^{\infty}e^{(-ikz^\prime))}dz^\prime))}=2\pi\delta(k)cos(kz)$

and that:

$lim_{k\rightarrow 0}\cos(kz)I_1(k\rho_<)K_1(k\rho_>))=\frac{\rho_<}{2\rho_>}$

we get:

$\mathbf{A}=\frac{\mu_0 n I\rho}{2}\hat\phi$ for $\rho < R$

and

$\mathbf{A}=\frac{\mu_0 n IR^2}{2\rho}\hat\phi$ for $\rho > R$

As expected.

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