How to find the magnetic field in a toroid with time-dependent current $I(t)$? The problem would be straightforward if the current $I$ were constant through the toroid. However, this is not the case.
The current is time dependent according to $I = I_{o} e^{-\gamma t}$.
So, if I'm right, a classic Ampere's Law approach doesn't work here. Because the $B$ field through the center of the toroid would change over time, it would cause an electric field, which must be taken into account in the magnetic field.
The toroid in question has a cross sectional radius $r$, internal radius $a$, and outer radius $b$. 
I feel like I am missing something, or approaching the problem incorrectly, as I do not know how to find the $E$ field induced by the changing $B$ field. From there, I do not actually know what the math would look like. I also have to calculate the self-inductance of the toroid after this, which I would do by finding the flux through the cross section of the toroid and comparing it to the current through the same area. But, I can't do that yet, because I don't have the magnetic field to find the flux!
 A: Ampere's law is an instantaneous relationship. It works "at time $t$". Simply don't worry about the time; treat it as a parameter. 
If the torus is huge or the current changes very quickly, the differential form or maybe the idea of retarded time might be used. 
A: You could use Jefimenko's equations to find the electric and magnetic fields due to the current.
First, there will be a magnetic field due to the current. But the retardation will make it so the magnetic field bleeds out of the torus instead of being perfectly contained. But just barely if the current does not change much in the time it take light to go from one side of the torus to the other.
Second, there will be a whole new magnetic field due to a new source term for the time rate of change of the current. And that source term will also produce an electric field. So the electric field has a cause, the time change of the current and that will also produce an additional magnetic field.
And in both cases, Jefimenko's equations provide the fields given the current as a function of time.
