Can Self-Induction and Mutual Induction Coexist Without Infinite Energy?

[Sefrez]

This may be a stupid question, but nonetheless it is confusing me (maybe I am stupid :p)

The equation states that there is an induced emf in a loop when there is a change in magnetic flux through that loop. Or V = -d$\phi$/dt

Though how does it work when you have self induction and induction created on another coil at which can then re-induce the one that started this loop? Is this even possible? If you had, say two solenoids, one in the other, then putting a varying current through the outer one would induce a current in the center one. If the outer coil's change in current is modeled linear as a function of time, then according to the equation, the induced emf in the center coil would be constant. This would mean that there is no run around change in flux back through the outer coil.

But what if the current through the outer coil was modeled by e^t, or even something like e^(2t)? Then, by the equation, the induced emf on the inner coil is increasing with time, and thus there is a change in flux though the outer coil which is what started the whole process. The "infinite'th" derivative of e^t is the same, and with e^(2t), it continuously increases.

When inducing this emf on the inner coil, it is in a direction opposing the change in current of the outer. But when it raps back around (inner inducing outer) it is in the same direction the current is already going.

I know this cannot be possible as energy would not be conserved.

So, my question is, what happens here? Do you need an infinite amount of energy to achieve re-induction of the outer coil or something?

Again, maybe this is a stupid question and I am not thinking clearly. It sounds like a bunch of bogus to me.

 

[jbsteele]

The answer to your question is no, you do not need an infinite amount of energy to achieve re-induction of the outer coil. The key to understanding this phenomenon is to realize that self-inductance always acts in opposition to the change in current. That is, when the current through the outer coil is increasing, the self-induced emf in the outer coil will be in the opposite direction to the increasing current, thus opposing the increase in current. The same is true for the inner coil; the self-induced emf will oppose the increase in current. This means that the two coils are essentially competing against each other in terms of how much current they can induce in the other. It is possible that one coil can win this competition if it has a larger self-inductance than the other, but this is unlikely. In most cases, the two coils will reach equilibrium at some point, where the self-induced emf in each coil is equal and opposite to the current in the other coil. In terms of energy, the two coils are exchanging energy back and forth. This causes the system to reach equilibrium without any external energy input.