# Energy transfer in electromagnetic induction

by entropy15
Tags: electromagnetic, energy, induction, transfer
 HW Helper P: 3,448 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: $B_x \hat{i} + B_y \hat{j} + B_z \hat{k}$ 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): $$\vec{E} = v \gamma (B_y \hat{i} - B_x \hat{j})$$ (where v is the absolute value of the speed, and I'm guessing you know what gamma is?) Also, the magnetic field is: $$\vec{B} = \gamma(B_x \hat{i} + B_y \hat{j}) + B_z \hat{k}$$ 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.
P: 37
 Quote by BruceW So his (implied) question is "where did the energy go?"
You got me right, Bruce. That was what I was trying to say.

 Quote by BruceW 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.
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.
Mentor
P: 17,322
 Quote by entropy15 But the energy of the wave emitted by the electromagnet (initially when it is switched on) is independent of this velocity.
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.
P: 37
 Quote by 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
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)
Mentor
P: 17,322
 Quote by entropy15 The total energy due to the radiation in all directions should be independent of velocity.
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.

 Quote by entropy15 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)
Due to relativistic effects a moving electromagnet will emit more total energy than an electromagnet at rest. Energy is frame variant.
P: 37
 Quote by DaleSpam Due to relativistic effects a moving electromagnet will emit more total energy than an electromagnet at rest. Energy is frame variant.

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?
Mentor
P: 17,322
 Quote by 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.
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.
P: 37
 Quote by DaleSpam Note that this is a very small effect for ordinary speeds. It is only significant at large fractions of c.
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?
 Mentor P: 17,322 Doppler. The total energy is not significantly affected for v<
P: 37
 Quote by DaleSpam Doppler. The total energy is not significantly affected for v<
What if we increase the number of turns in the coil? Would that not mean the coil acquiring more energy?
Mentor
P: 17,322
 Quote by entropy15 What if we increase the number of turns in the coil? Would that not mean the coil acquiring more energy?
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).
P: 37
 Quote by 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).
Sorry I am unable to think of any reason here. Could you please explain
 Mentor P: 17,322 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.
P: 37
 Quote by 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..
I was referring to the turns in the absorbing coil placed at a distance x from the electromagnet.
 Mentor P: 17,322 Me too. Think about the field generated by the induced current in the first turn. How does that affect the total field seen by the second turn.
P: 37
 Quote by DaleSpam Doppler. The total energy is not significantly affected for v<
The point I was trying to make was that the initail energy due to the current induced in the coil is entirely due to the electromagnetic wave.
The kinetic energy of the electromagnet cannot contribute to the induced current, as it does not decrease initailly.

Lets consider the time interval between x/c and when the coil begins to feel resistance due to mutaul induction. It will be less than 2x/c since the electromagnet is moving towards the coil.
The energy due to the current in the coil during this time cannot be greater than the energy in the electromagnetic wave intially radiated.

But if we increase the value of v, the energy in the coil increases because of a larger change in flux. But there is no noticeable increase in the radiation energy. (v<<c)

If we consider the frame of the moving electromagnet there is no Doppler effect.
All the electromagnet sees is the coil moving towards it.
Here again we can see that the energy in the coil (between x/c and 2x/c) increases with increase in the relative velocity.
PF Gold
P: 1,165
 But there is no noticeable increase in the radiation energy. (v<
If the electromagnet radiates isotropically in its frame (if it has symmetric shape), the energy radiated into all directions per unit time is Lorentz invariant; it is the same in all frames. Let's say the electromagnet radiated 1 J in one second, in its own frame of reference.

What is important, is that in the frame of the coil, the electromagnet is moving towards it. When a source of isotropic radiation moves in some direction, the radiation is released preferentially to that direction. Check

The bunches of charged particles circling in synchrotron move so fast that the radiation is needle-like, similar to laser, only much brighter and not monochromatic.

With the electromagnet, it is similar; even if it moves slowly, there is more radiation going to the coil than in the other directions.

As the velocity is increased, coil receives greater and greater power. However, there is a limit: when v approaches c, the coil receives almost all the radiated power 1 J/s and this is the maximum. Of course, as the processes in the source are slowed down (dilatation) , it will receive it for a long time and thus the net amount of energy received in the end can be much greater than 1 J.

Where did the extra energy came from? From the total energy of the electromagnet; as the net energy of the electromagnet decreases by radiation, in the frame of the coil the electromagnet loses also momentum via loss of its mas (the velocity is unaffected).
Mentor
P: 17,322
 Quote by entropy15 The point I was trying to make was that the initail energy due to the current induced in the coil is entirely due to the electromagnetic wave. The kinetic energy of the electromagnet cannot contribute to the induced current, as it does not decrease initailly.
The EM wave is what carries the KE away. The KE of the magnet does decrease as soon as the electromagnet radiates. Remember, the KE decreases due to the loss of mass from radiating energy, even if the velocity remains constant.

 Quote by entropy15 Lets consider the time interval between x/c and when the coil begins to feel resistance due to mutaul induction. It will be less than 2x/c since the electromagnet is moving towards the coil. The energy due to the current in the coil during this time cannot be greater than the energy in the electromagnetic wave intially radiated.
Yes, that is not in doubt at all. The point is that the energy in the EM wave depends on the reference frame. In reference frames where the magnet was initially moving the energy in the EM wave is greater than in the frame where it was stationary. Energy is frame variant.

 Quote by entropy15 But if we increase the value of v, the energy in the coil increases because of a larger change in flux. But there is no noticeable increase in the radiation energy. (v<
No noticeable increase in the TOTAL radiation energy, but there is a noticeable increase in the energy through the coil. Doppler.

 Quote by entropy15 If we consider the frame of the moving electromagnet there is no Doppler effect.
This is incorrect, the Doppler effect depends only on the relative velocity. In any frame there is the exact same amount of Doppler effect. In the magnet's frame, of course, the Doppler effect is due entirely to the movement of the loop.

 Quote by entropy15 All the electromagnet sees is the coil moving towards it. Here again we can see that the energy in the coil (between x/c and 2x/c) increases with increase in the relative velocity.
Doppler.

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