# Correctness of the quasistatic EMFT approximation

1. Jan 9, 2016

### TubbaBlubba

Hello! I'm just trying to make sure I understand Maxwell, Biot-Savart, Faraday, etc well enough. What I'm wondering about is when the quasistatic treatment is indeed approximate, and when it is, assuming an equal if time-varying current, correct.

Suppose, first, that we have a steadily increasing current I(t) - that is, dI/dt = k, some constant. The wire (or solenoid, or whatever) is short enough that the current is equal everywhere. We compute the magentic field from Biot-Savart, find the change over time in flux, and thus obtain the induced emf by Faraday's law of induction. As I understand the textbook explanation, this is only approximate (however close), because Biot-Savart only applies for steady current. But any error, under our assumptions, could only come from the time derivative of E in the Ampere-Maxwell law. Since our induced E is constant (because dI/dt is constant) this term is zero. Similarly for the self-induced countercurrent - it is at most a constant term that disappears when we take the time derivative of B.

Thus it appears to me that the quasistatic approximation to the emf induced by a steadily increasing current is only approximate if the wire cannot be assumed to be electrically neutral - then we have charges piling up resulting in a time-dependent E-field and so on. The approximation to the magnetic field will be off by a constant term, the self-induced current.

But on the other hand, if we have a sinusoidal current, even IF the wire is taken to be neutral, the curl of E will still have a time-dependent component, thus a non-zero time derivative in the Ampere-Maxwell equation and the self-induced current, and so on to the nth time derivative until the 1/c term in the curl of B makes it negligible.

Is this theoretically correct, or is there some other time-independent assumption I'm neglecting?

Thanks.

2. Jan 14, 2016