electromagnetism - Toroid with variable current and Ampere's Law

Say you have a current $I(t)$ (notice the time dependence) flowing through a Toroid with $N$ total loops and all the usual approximations: $(b-a) \ll r,\; B=0$ outside.

_{(source: gsu.edu)}

You are asked to calculate the magnetic field $B$. Could you apply Ampere's Law to obtain the familiar form of $B$ found for example here?:

$$B~=~\frac{\mu N I}{2\Pi r}.$$

Answer

You will in general also need to include the term from the time varying electric field. In integral form:

$$ \oint_{\partial A}{\vec{B}\cdot d\vec{s}} = \mu_0I + \frac{1}{c^2}\frac{\partial}{\partial t}\int_{A}\vec{E}\cdot dA $$

If $I(t)$ changes slowly so that the second term is small, you can do the calculation as in the static case.

For any given application, it's important to examine the approximations you are making, and determine if they are justified in your particular case. Here, you should also realize that Ampere's Law is telling you the component of B along the path of integration. Inside the toroid, there is a radial component as well.