Magnetic Field of A Toroid

1. Dec 15, 2013

richyw

1. The problem statement, all variables and given/known data

I'm working through a proof that says the magnetic field of a toroid is circumferential at all points inside and outside the toroid. I can follow most of the proof, but am a bit confused where the first equation comes from.

Here is the figure from the textbook (Griffith's 4th Ed).

http://media.newschoolers.com/uploads/images/17/00/67/76/17/677617.jpeg [Broken]

To begin the proof, Griffiths starts with the field at $\mathbf{r}$ due to the current element at $\mathbf{r}'$.$$d\mathbf{B}=\frac{\mu_0}{4\pi}\frac{\mathbf{I}\times\mathbf{\hat{{u}}}}{u^3}dl'$$

2. Relevant equations

$$\mathbf{B}=\frac{\mu_0}{4 \pi}\int\frac{\mathbf{I}\times\mathbf{\hat{u}}}{u^2}dl'$$

3. The attempt at a solution

I'm just confused at how Griffiths got from the Biot-Savart law above, into the equation he posted in the question. (I replaced the script r with a u). Where does the cubed term come from?

Last edited by a moderator: May 6, 2017
2. Dec 15, 2013

xophergrunge

I believe that is a mistake in the text. I think he meant it to be the vector r and not r hat, in which case the denominator would be cubed.

3. Dec 16, 2013

rude man

I agree with post #2. Otherwise the dimensions don't check out.

4. Dec 16, 2013

richyw

cool thanks. I was thinking that, but I wanted to confirm!

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted