Let’s repeat what we did to come up with Gauss law, but this time on the magnetic field.
As a first step, though, I want to finally get a good definition of “current density”: it’s a vector field that consists of a charge density and a velocity vector , each of which is a function of space. In our example of an infinite line current, this density was concentrated along the -axis, where the velocity was vertical. But it could exist along a surface, or throughout space; a single particle of charge moving with velocity is a current density concentrated at a single point.
Anyway, so the Biot-Savart law says that the differential contribution to the magnetic field at a point from the current density at point is
So, as for the electric field, we want to integrate over :
Last time we spent a while noting that the fraction here is secretly a closed -form in disguise, so its divergence is zero. This time, I say it’s actually a conservative vector field:
Indeed, this is pretty straightforward to check by rote calculation of derivatives, and I’d rather not get into it. The upshot is we can write:
where the extra term on the second line is automatically zero because the curl is in terms of and the current density depends only on . I write it in this form because now it looks like the other end of a product rule:
Indeed, this is clearer if we write it in terms of differential forms; since the exterior derivative is a derivation we can write
for a function and a -form . If we flip over to a vector field this looks like
Okay, so now we see that is the curl of some vector field, and so the divergence of a curl is automatically zero:
Coupling this with the divergence theorem like last time, we conclude that there is no magnetic equivalent of “charge”, or else the outward flow of through a closed surface would be the integral on the inside of such a charge. But instead we find