Energy transfer in electromagnetic induction

entropy15

Yes, I can see that now. Thanks much.

But what if we repeat the original experiment of the magnet and coil by replacing the permanent magnet with this electromagnet.

Initially the electromagnet is turned off. Then it is nudged so that it moves at a constant velocity v.
Now we power on the electromagnet at instant t=t1.
When will the current be induced in the coil (which is at a distance x) and when will the electromagnet experience resistance?

Would the results be similar to what we get when we had the permanent magnet there or is it different with an electromagnet?

DaleSpam

What do you think? What would you expect to be similar and what would you expect to be different in this case?

BruceW

I am impressed by entropy15's perseverance at trying to understand this problem fully :) The point I think he is trying to make is that when the electromagnet gets switched on, there is a delay of 2x/c before it feels the resistance due to mutual inductance (or slightly less, since it is moving at some constant velocity). But the current is induced in the coil after a delay of only x/c. So his (implied) question is "where did the energy go?"

The problem is most simple when the magnet and coil are stationary with respect to each other (and then we can do a Lorentz transform to find the answer for when the magnet is moving at constant velocity WRT the coil). So for now, I will assume the electromagnet is stationary WRT the coil. Now, when the electromagnet is switched on, a wave will be emitted, immediately carrying energy away from the magnet (this doesn't care if the coil is there or not). And if the coil is there, some of the energy will be taken out of the EM field to start moving the charges around the coil.

Now if the electromagnet were moving with some constant velocity WRT the coil, then a similar thing will happen when the switch is flicked on. (i.e. the electromagnet emits a wave of energy). And after a short time, the current through the electromagnet will become approximately constant. So after the delay of x/c, the coil will feel a brief EM wave, then a steady magnetic field that is slowly increasing because the magnet is getting closer to the coil. So after some time, the electromagnet will start feeling resistance due to mutual inductance. From this explanation, there is no problem that initially the coil has an induced current, while the magnet feels no resistance from mutual inductance, because initially, when the switch was turned on, the electromagnet released some energy into the EM field. So the situation satisfies energy conservation.

Edit: WRT means "with respect to" and EM means "electromagnetic". Also I should say, my whole explanation hinges on the assumption that a stationary electromagnet will emit an electromagnetic wave when it is first switched on. I am pretty sure this is right.

BruceW

Also, entropy15, you mentioned on the last page about relativity, and how the EM field looks different in different inertial frames. I don't think it has much relevance to the problem we are talking about, but I decided to work out the EM field due to a magnet moving at constant velocity. (This is when there are no other coils, or any other EM fields, apart from that created by the magnet). (Also, I am assuming that in the rest frame of the magnet, there is zero electric field). Let the magnetic field in the rest frame of the magnet be: [itex]B_x \hat{i} + B_y \hat{j} + B_z \hat{k} [/itex] then, in a reference frame moving to the left WRT the rest frame (i.e. according to an observer who sees the magnet moving to the right):
[tex]\vec{E} = v \gamma (B_y \hat{i} - B_x \hat{j}) [/tex]
(where v is the absolute value of the speed, and I'm guessing you know what gamma is?) Also, the magnetic field is:
[tex]\vec{B} = \gamma(B_x \hat{i} + B_y \hat{j}) + B_z \hat{k} [/tex]
So (assuming that I calculated correctly), even though there is zero electric field in the rest frame, there is a non-zero electric field in this frame where the magnet is moving at constant velocity. Also, the magnetic field has been 'stretched' in both directions perpendicular to the direction of motion. But the magnetic field in the direction of motion remains unchanged.

Aaanyway, as I said, I don't think these equations are much use to the problem we are talking about.

entropy15

You got me right, Bruce. That was what I was trying to say.

When the electromagnet is switched on, a magnetic field is also set up around it.
At time x/c the effect of this magnetic field reaches the coil. - (since nothing travels faster than light)

Now since the electromagnet is moving at a constant velocity v, there would be change in the magnetic flux linking the coil. Hence there would also be a current induced.

Now I think we can say that the change in the flux linking the coil would be more if the electromagnet was moving more fast. Hence more the induced current.

So lets see what happens between the time interval x/c and the time the electromagnet faces resistance due to mutual induction. This will be less
than 2x/c since it is moving towards the coil.

If the electromagnet was moving with a large velocity we can expect a large change in flux and hence the current induced.

But the energy of the wave emitted by the electromagnet (initially when it is switched on) is independent of this velocity. So how does it account for the large current induced in the coil.

I believe that energy is always conserved. The only thought was that the initial resistance faced by the electromagnet (as soon as it is switched on )was dependent on
whether there is any coil in the vicinity.

DaleSpam

This is not correct. I am not certain why you would think this, but it is wrong. Not only is it dependent on the velocity, it is also dependent on the angle of approach. This is called the Doppler effect (see http://en.wikipedia.org/wiki/Relativ...Doppler_effect). In the forward direction the wave will be blue-shifted and therefore have a higher energy than in the reverse direction where it will be red-shifted. Thus the energy of the wave is higher in the region where the change in flux is higher.

entropy15

The total energy due to the radiation in all directions should be independent of velocity.
Isn't that so? Otherwise an electromagnet moving at a non zero velocity will emit more than an electromagnet at rest. (when they are switched on)

DaleSpam

The total energy due to the radiation in all directions is not relevant here, only the energy in the direction of the loop, which is higher.

Due to relativistic effects a moving electromagnet will emit more total energy than an electromagnet at rest. Energy is frame variant.

entropy15

The energy in the radiation should be coming from the source driving the electromagnet.
Assume that the electromagnet is powered by a power source - a battery or a charged capacitor.

Now if the electromagnet is moving more and more faster (at a constant velocity) does it mean that the source has to provide more and more energy to power on the electromagnet?

DaleSpam

The energy in the radiation also comes from the KE of the source. When a capacitor is discharged to power the magnet then by E=mc² that capacitor has less mass. So in a frame where it is moving it also has less KE. So not only is the electrical potential energy in the capacitor decreased, but also the KE of the capacitor is decreased. That additional energy goes into the radiation.*

Note that this is a very small effect for ordinary speeds. It is only significant at large fractions of c.

*this explanation is a little sloppy, a better explanation would be in terms of the four-momentum. If you are interested, please open a new thread in the relativity sub-forum.

entropy15

So for smaller velocities (compared to c) there should be no noticeable increase in the amount of energy being emitted by the electromagnet whether it is stationary or moving.

So assume that the velocity of the electromagnet involved in the experiment I mentioned earlier (post 56) is small compared to c.
But it is moving so as to cause a significant change in the flux linking the coil. Then how do we we explain it?

DaleSpam

Doppler. The total energy is not significantly affected for v<<c, but it is still concentrated in the forward direction. As I said in 59 above, the total energy is not terribly important in this scenario, only the energy at the loop.

entropy15

What if we increase the number of turns in the coil? Would that not mean the coil acquiring more energy?

DaleSpam

What do you think? Try to reason this from what you know of Maxwell's equations, especially the fact that energy is conserved in them and the fact that they are linear (superposition).

entropy15

Sorry I am unable to think of any reason here. Could you please explain

DaleSpam

Superposition means that if you have two sources then the total field is the sum of the field from each of the two individual sources. Think how that might apply to increasing the number of turns.

entropy15

I was referring to the turns in the absorbing coil placed at a distance x from the electromagnet.