Question

Having some trouble with the Biot-Savart Law.

Answer 1

So I have with toroid (see figure). I'm working through an example that proves the magnetic field of a toroid is circumferential at all points.

I know that the Biot-Savart law says \[\mathbf{B}=\frac{\mu_0}{4\pi}\int\frac{\mathbf{I}\times\mathbf{\hat{u}}}{u^2}dl'\]Where \(\mathbf{u}=\mathbf{r}-\mathbf{r}'\) is the separation vector (cursive r in the figure). Griffiths then says that the biot Savart law gives for this case\[d\mathbf{B}=\frac{\mu_0}{4\pi}\frac{\mathbf{I}\times\mathbf{\hat{u}}}{u^3}dl'\]Where does this come from?

Answer 2

i think he just derive the integral

Answer 3

I don't understand why \(u^2\rightarrow u^3\) in the denominator. If you take the derivative of the integral, shouldn't it just be like "removing the integral sign"?

Answer 4

i think so too

Answer 5

@experimentX

Answer 6

Remember the first form above is where r is the unit vector acting towards a point.

So, for a vector the unit vector is defined as a vector divided by its magnitude as such
the unit vector will be the full displacement vector divided by its magnitude.

r^ = r/ IrI

So that's where the u^3 comes from

Hope this helps :)

Answer 7

The u^3 in the differential is a misprint. It should be u^2 as was originally surmised.