Why doesn't current induced by changes in flux affect the flux, while current induced by a battery does?

Question

Faraday's law states $V=-\frac{\mathrm{d}F}{\mathrm{d}t}$, where $F$ is flux. Since $V=IR$ for passive circuits, $I = V/R = -\frac{1}{R}\frac{\mathrm{d}F}{\mathrm{d}t}.$

But by the Biot-Savart law, the induced current should produce a magnetic field, which should change the magnetic flux within the loop, and, therefore, change the current. However, my textbook implies (by not addressing it) that this doesn't occur.

We use this sort of feedback to explain inductance and back-emf, but why do those laws not also apply when the voltage difference is induced by flux?

There's a (likely) chance that I've just misunderstood some concept or that this question doesn't make sense for some reason.

Answer 1

Faraday's law states that $V = - \frac{dF}{dt}$ where $F$ is the overall magnetic flux due to all sources through the area bounded by the loop. As you point out, it is precisely because of this that inductance etc. make sense.

You may have likely encountered examples in your textbook where the rate of change of magnetic flux due to an external source is constant i.e. $\frac{d^2F_{ext}}{dt^2} = 0$. In this case, as the contribution due to the induced current may be expressed as $$F_{curr} = k\frac{dF}{dt} = k\frac{dF_{ext}}{dt} + k\frac{dF_{curr}}{dt}$$ we have that $\forall n \ge 1, \frac{d^nF_{curr}}{dt^n} = k \frac{d^{n+1}F_{curr}}{dt^{n+1}}$, which for 'well behaved' changes implies $\frac{dF_{curr}}{dt} = 0$ i.e. $$\frac{dF}{dt} = \frac{dF_{ext}}{dt}$$ To summarize, for a flux with a constant rate of change, the induced current is constant, and therefore does not influence the emf across the loop.

Answer 2

Why doesn't current induced by changes in flux affect the flux, while current induced by a battery does?

Because electromagnetism features a "screw" mechanism. Take a look at Minkowski’s Space and Time:

"In the description of the field caused by the electron itself, then it will appear that the division of the field into electric and magnetic forces is a relative one with respect to the time-axis assumed; the two forces considered together can most vividly be described by a certain analogy to the force-screw in mechanics; the analogy is, however, imperfect".

Maxwell said something similar in On Physical Lines of Force:

"A motion of translation along an axis cannot produce a rotation about that axis unless it meets with some special mechanism, like that of a screw."

See the electromagnetic field article on Wikipedia and note where it says this: "Over time, it was realized that the electric and magnetic fields are better thought of as two parts of a greater whole — the electromagnetic field." This is what Jackson was indirectly referring to in section 11.10 of his Classical Electrodynamics where he said "one should properly speak of the electromagnetic field Fμν rather than E or B separately". IMHO contemporary teaching doesn't emphasize this enough, and instead spends a lot of time talking about electric fields and magnetic fields as if they're two totally different things. When you push a current up a wire, you don't create a magnetic field, you merely reveal an aspect of the "greater whole" electromagnetic field. Think of a pump-action screwdriver: you push forward, and the result is a rotation. Then think of an ordinary screwdriver: you rotate the handle, the screw pushes forward, but that doesn't make the handle rotate. The electromagnetic field is akin to this, which is why we have the right hand rule for both the current in the wire, and for screw threads.