I've been trying to follow this post on deriving the Biot-Savart law from Maxwell's Equations but am getting stuck on this step:
$$-\frac{\mu_0}{4\pi}\iiint{\nabla\times\frac{J}{|r-r'|} d^3r'}.$$
Mainly the curl of current density part which would expand to:
$$\nabla\times\frac{J}{|r-r'|}=\frac{\nabla\times J}{|r-r'|}+\nabla\frac{1}{|r-r'|}\times J$$
The only way this would reduce to $\frac{|r-r'|\times J}{|r-r'|^3}$ would be if $\nabla\times J=0$.
So I tried showing that $\nabla\times J = 0$ in a steady-state situation, however, I'm getting stuck.
Assumption: Steady-state current which implies that for all of space and time $\frac{\partial B}{\partial t}=0$
We start with Ampere's law,
$$\nabla\times B=\mu_0J+\mu_0\epsilon_0\frac{\partial E}{\partial t}$$
Curl of both sides
$$-\nabla^2B=\mu_0\nabla\times J+\mu_0\epsilon_0\frac{\partial(\nabla\times E)}{\partial t}$$
And because $\nabla\times E = -\frac{\partial B}{\partial t}$
$$-\nabla^2B=\mu_0\nabla\times J-\mu_0\epsilon_0\frac{\partial^2 B}{\partial t^2}$$
$$\mu_0\epsilon_0\frac{\partial^2 B}{\partial t^2}=\mu_0\nabla\times J+\nabla^2B$$
However, this is where I'm getting stuck, there are two scenarios for $\partial^2 B / \partial t^2 = 0$: Either $\nabla^2 B$ is zero meaning that $\nabla\times J$ would also be zero, or that $\nabla^2 B$ is non-zero meaning that $\nabla\times J=-\nabla^2 B$.
So now I would need to show that $\nabla^2 B=0$ which I'm not sure is true (For example, wouldn't it be $\infty$ at the wire – assuming thin wire?)
So my question is: is $\nabla\times J=0$ in steady-state? If so, where am I going wrong in my proof?
Best Answer
This is a consequence of bad notation.
In writing the formula:
$$\nabla × \vec{A} = \vec{B}$$
You need to be very careful about which variables you are taking the curl of.
You should write:
$$\nabla_{\vec{r}} × \vec{A}(\vec{r}) = \vec{B}(\vec{r})$$
This is because we are taking the curl with respect to the variables $\vec{r} = x\hat i + y\hat j + z\hat k$
When we apply the $\nabla_{\vec{r}} ×$ operator on $\vec{J}$ we need to keep track of what $\vec{J}$ is a function of.
Your confusion lies in the fact that yes, in maxwells equations, $\vec{J}$ and $\vec{B}$ are both functions of $\vec{r}$.
However, when solving poissons equation, $\vec{J}(\vec{r})$ changes to $\vec{J}(\vec{r}')$
Meaning,
$\nabla_{\vec{r}} × \vec{J}(\vec{r}') = 0$
As $\vec{J}(\vec{r}')$ is independant on $\vec{r}$
So your conclusions you've drawn are presumably correct if $\nabla_{\vec{r}'} × \vec{J}(\vec{r}') = 0$, but that's not what we are saying when deriving biot savart.