# 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.

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?:

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

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$$\oint_{\partial A}{\vec{B}\cdot d\vec{s}} = \mu_0I + \frac{1}{c^2}\frac{\partial}{\partial t}\int_{A}\vec{E}\cdot dA$$
If $I(t)$ changes slowly so that the second term is small, you can do the calculation as in the static case.