Energy gained from induced current

[ELB27]

I have a confusion about the energy effects of current induction. Suppose I place a coil of wire near a long, high-voltage AC power-line. The changing current and hence, the changing magnetic field it produces will induce current in my coil. Then, I can connect the coil to a resistor and get heat energy out of the system, apparently for nothing. I highly suspect that the induced current in my coil will slow down the current in the power-line, but how exactly? I know about mutual inductance, but if I understand correctly the energy lost in this process of "back emf" is retrievable whereas the energy lost via Joule heating is not. (assuming that the wire is infinite for instance).

Moreover, if I put my (say, circular) coil in a bigger loop of increasing current, a current will be induced in my coil such that its flux will oppose the change in the original one. Let's say that the original flux was "positive". Then, there will be an increasing flux in the opposite, negative direction through the bigger loop, causing a current producing a positive flux to increase even further! Clearly, this is absurd, yet I can't figure out exactly where is the caveat. My suspicion here is that I forget about the flux of the loop through itself and that if I look at the total flux, it stays such that its change is equal to the source emf or constant.

Can anyone settle my confusion?
Any comments will be greatly appreciated!

 

[Jano L.]

I highly suspect that the induced current in my coil will slow down the current in the power-line, but how exactly? I know about mutual inductance, but if I understand correctly the energy lost in this process of "back emf" is retrievable whereas the energy lost via Joule heating is not. (assuming that the wire is infinite for instance).

Your suspicion is correct, your coil will act back on the AC line. If there is no resistor, most of the energy will go where it went before, the coil will just change the distribution of energy and its flux locally around the coil.

If there is a resistor, it will change the energy flux more significantly and there will be positive flux of EM energy going to the resistor from all directions.

Moreover, if I put my (say, circular) coil in a bigger loop of increasing current, a current will be induced in my coil such that its flux will oppose the change in the original one. Let's say that the original flux was "positive". Then, there will be an increasing flux in the opposite, negative direction through the bigger loop, causing a current producing a positive flux to increase even further!

The part in italics is not right, because the increasing current producing the positive flux experiences also much stronger self-damping effect (self-inductance of the loop, or, more accurately action of parts of the loop on each other).

If, hypothetically, the effect of the small coil on the current of the loop would be stronger than the self-damping effect, both loops would drive each other and increase their current for free indefinitely. If somebody could achieve this, it would violate the laws of electromagnetism.

 

[ELB27]

The part in italics is not right, because the increasing current producing the positive flux experiences also much stronger self-damping effect (self-inductance of the loop, or, more accurately action of parts of the loop on each other).

If, hypothetically, the effect of the small coil on the current of the loop would be stronger than the self-damping effect, both loops would drive each other and increase their current for free indefinitely. If somebody could achieve this, it would violate the laws of electromagnetism.

So it all hinges on the mutual inductance being smaller than the self-inductance? Both are geometric quantities, so can it be proved that regardless of the size and shape of the loops, their mutual inductance must be smaller than each individual's self-inductance?

Your suspicion is correct, your coil will act back on the AC line. If there is no resistor, most of the energy will go where it went before, the coil will just change the distribution of energy and its flux locally around the coil.
If there is a resistor, it will change the energy flux more significantly and there will be positive flux of EM energy going to the resistor from all directions.

Interesting. What is the mechanism through which it happens? I.e., how would I get a more rigorous feeling of this situation - flux change, Lorentz force, Poynting Theorem? Also, I thought that the presence of a resistance-less coil would be equivalent to its absence altogether - how does it change the energy distribution?

 

[Dale]

While there certainly is some mutual inductance and some "feedback" to the primary wire, that can be made very small in certain circumstances, so it is not a universal explanation.

The universal explanation is Poynting's theorem. It shows that any system which obeys Maxwell's equations will conserve energy. The details are only important as a kind of homework exercise about how to apply the theorem to a particular setup.

 

[Jano L.]

So it all hinges on the mutual inductance being smaller than the self-inductance?

The important quantities for comparison are the magnitudes of induced emf-s due to the small coil (with current $i$) and the loop (current $I$) in both the small coil and the loop.

To sustain self-supplied increase of both currents, the emf in one loop due to another would have to be stronger than the self-induced emf. Mathematically, both inequalities

$$
M\frac{dI}{dt} > -l\frac{di}{dt}
$$

$$
-M\frac{di}{dt} > L\frac{dI}{dt}
$$

would have to be satisfied. These imply

$$
M > \sqrt{lL}
$$

as a necessary consequence. However, mutual inductance $M$ cannot be higher than $\sqrt{lL}$, for it attains maximum when the two loops are in close contact, and then the coils almost coincide so $M$ is slightly lower than $L=l$. If the two coils could somehow occupy the same space, all three numbers would be the same, but there is no way to get the two coils closer than that.

Interesting. What is the mechanism through which it happens? I.e., how would I get a more rigorous feeling of this situation - flux change, Lorentz force, Poynting Theorem?

To get a feeling for EM phenomena you need to study electromagnetic phenomena by theory and experiments a lot. Good books that are quite easy to get are Panofsky&Phillips, Purcell, Feynman, Landau&Lifshitz. These are books by physicists; also many engineering books have a useful things to add.

Also, I thought that the presence of a resistance-less coil would be equivalent to its absence altogether - how does it change the energy distribution?

No, ideal coil would enable presence of electric current and necessitate a change of EM field in its vicinity.

 

[ELB27]

Thank you for the detailed answer! Perhaps I will fiddle around a bit more with the equations in a simplified scenario to iron this one. My main concerns are satisfied.

 

[anorlunda]

I have a confusion about the energy effects of current induction. Suppose I place a coil of wire near a long, high-voltage AC power-line.

Read the Wikipedia article on transformers. http://en.m.wikipedia.org/wiki/Transformers

You may not realize it, but the in your scenario you are creating a transformer. All the equations for transformers may be applicable.

 

[ELB27]

Read the Wikipedia article on transformers. http://en.m.wikipedia.org/wiki/Transformers

You may not realize it, but the in your scenario you are creating a transformer. All the equations for transformers may be applicable.

Thank you! An interesting parallel. The only problem is that the flux through the two loops is quite different. Other than that it does seem to be similar.

 

[anorlunda]

The only problem is that the flux through the two loops is quite different. Other than that it does seem to be similar.

Not a big problem. Leakage flux refers to flux through one winding thst does not go through the other winding. In a regular transformer, leakage flux is small, but in this case it is large. The transformer equations still apply.