Correctness of the quasistatic EMFT approximation

[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.

 

[eljose79]

Thank you for your question about the quasistatic approximation in electromagnetic theory. You have raised some interesting points and I will do my best to address them.

Firstly, it is important to note that the quasistatic approximation is indeed an approximation and is not exact. It is used in situations where the time variations are small compared to the characteristic time scale of the system. In your example, the time variation of the current is small compared to the characteristic time scale of the system (i.e. the time it takes for the electromagnetic fields to propagate through the system). This allows us to neglect the time derivatives in the equations and simplify the analysis.

In the case of a steadily increasing current, as you have correctly pointed out, the approximation is only valid if the wire can be assumed to be electrically neutral. This means that there are no charges building up in the wire, which would result in a time-dependent electric field. If this assumption is not valid, then the quasistatic approximation will not be accurate and the induced emf will be affected.

In the case of a sinusoidal current, the quasistatic approximation is still valid as long as the time variations are small compared to the characteristic time scale of the system. However, as you have also noted, the time derivative of the electric field will be non-zero, which will result in a self-induced current. This is accounted for in the Ampere-Maxwell law, where the time derivative of the electric field is included in the curl of the magnetic field. As long as the time variations are small, this term will be negligible and the quasistatic approximation will still hold.

In summary, the quasistatic approximation is only valid in situations where the time variations are small compared to the characteristic time scale of the system. If this is not the case, then the approximation will not be accurate and the induced emf will be affected. I hope this helps to clarify your understanding of the quasistatic approximation. If you have any further questions, please do not hesitate to ask.