Energy transfer in electromagnetic induction

[Drakkith]

Next the electromagnet is switched on so that it produces a magnetic field. Now since it is moving, it should try putting energy to the electromagnetic field.

So won't this oppose the current flowing in the electromagnet itself? trying to decrease the magnetic field.

No, from the frame of the electromagnet it is stationary and it has no field built up prior. When you had the regular magnet, you also had a magnetic field built up which has its own frame of reference. When the magnet was bumped the two frames were no longer stationary with each other and the movement caused the field to oppose the acceleration and movement for a moment.

 

[entropy15]

No, from the frame of the electromagnet it is stationary and it has no field built up prior. When you had the regular magnet, you also had a magnetic field built up which has its own frame of reference. When the magnet was bumped the two frames were no longer stationary with each other and the movement caused the field to oppose the acceleration and movement for a moment.

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?

 

[Dale]

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: 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.

 

[entropy15]

So his (implied) question is "where did the energy go?"

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

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 let's 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.

 

[Dale]

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/Relativistic_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]

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)

 

[Dale]

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.

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.

 

[entropy15]

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?

 

[Dale]

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.

 

[entropy15]

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?

 

[Dale]

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]

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.

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

 

[Dale]

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).

 

[entropy15]

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

 

[Dale]

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]

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.

 

[Dale]

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.

 

[entropy15]

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.

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.

 

[Jano L.]

But there is no noticeable increase in the radiation energy. (v<<c)

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

https://en.wikipedia.org/wiki/Synchrotron_radiation

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).

 

[Dale]

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.

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.

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)

No noticeable increase in the TOTAL radiation energy, but there is a noticeable increase in the energy through the coil. Doppler.

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.

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.

 

[entropy15]

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..

You mean to say that there is limit to the energy that can be transferred to the coil between the interval x/c and 2x/c.

Can we not increase the energy transferred initially by increasing the number of turns in the coil. More turns mean more current flowing in the coil.

 

[BruceW]

This goes back to the point before. I think DaleSpam's answer was: "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. My hint is that the equation:
\displaystyle{\varepsilon}=-N \frac{d \Phi}{dt}
Uses a lot of assumptions, and if you break those assumptions, you cannot expect the equation to give correct results.

Edit: actually, it doesn't use a lot of assumptions, but the simple case of increasing number of turns to increase the current through the coil does introduce assumptions.

Another Edit: and generally, it is assumptions used along with this equation that have caused the problems in this thread. For example, the assumption "that the magnetic field at the magnet and at the coil is approximately the same" is often used with this equation, but this assumption becomes false when the magnet and coil are far away from each other.

 

[Dale]

You mean to say that there is limit to the energy that can be transferred to the coil between the interval x/c and 2x/c.

Can we not increase the energy transferred initially by increasing the number of turns in the coil.

In addition to BruceW's point above, I would like to point out that the two statements are not mutually exclusive. Specifically, it is possible that the statements "there is a limit to the energy that can be transferred" and "we can increase the energy transferred by increasing the number of turns" can both be true.

 

[entropy15]

In addition to BruceW's point above, I would like to point out that the two statements are not mutually exclusive. Specifically, it is possible that the statements "there is a limit to the energy that can be transferred" and "we can increase the energy transferred by increasing the number of turns" can both be true.

How is that possible?

 

[Dale]

How is that possible?

Like this (ignore the scale and units of the vertical axis, this is just to give a general idea of the concept of a monotonically increasing function with a horizontal asymptote)

If each dot represents the energy extracted by a coil with i turns then it is both true that "there is a limit to the energy that can be transferred" and "we can increase the energy transferred by increasing the number of turns".

 

[entropy15]

Like this (ignore the scale and units of the vertical axis, this is just to give a general idea of the concept of a monotonically increasing function with a horizontal asymptote)

Thanks for clarifying. This seems to indicate that the amount of energy transferred converges to a fixed value as N (no. of turns) tends to infinity.

Is this because the successive turns in the coil are linked to lesser magnetic flux?

 

[Dale]

Yes, it is because each turn reduces the flux seen by the other turns. This can be seen through the superposition principle.

Suppose that you have two turns, consider them to be two separate loops. There is a current in loop A which creates a field which opposes the change in the external field. By superposition the field seen by loop B is the sum of the field from loop A and the external field, which is less than the change in the external field. So the current induced in B is a function of the external field and the current induced in A where current in A reduces the induced current in B.

Then, to consider the loops as separate turns in a single coil simply equate the current in A to the current in B.