Energy transfer in electromagnetic induction

[entropy15]

What do you mean the energy stored in the field is fixed? If we quickly accelerate to a high speed it will have more energy stored in the field than simply pushing the magnet slowly into the coil. (From what it looks like to me)

If the magnet is stationary then the magnetic field around it is constant and so should the energy stored in it.
On accelerating if the energy stored is increasing it is due to the fact that the kinetic energy of the magnet being converted to the energy of the magnetic field.

 

[BruceW]

So what is the actual delay? 0 or (x/c) or (2x/c) ?
If it is x/c there will be no problem with energy.
But the point here is how can the delay be x/c since the resistance is caused by the induced current in the coil and there should be a time gap between the cause and effect since they are physically separated by a distance x.

I would think the delay is 2x/c since nudging the magnet would cause a wave, which travels to the coil, then the acceleration of charges in the coil causes a wave which travels back. And as Drakkith says, I don't see a problem with energy since the energy can get stored in the EM field. Also, I think you are right that it needs to be at least 2x/c because of the causality argument.

 

[BruceW]

If we consider the coil to be moving and the magnet stationary, then it is pretty straightforward that the coil will experience resistance instantaneously.This is because the magnetic field is present where the coil is.

There's two things going on 1) the 'nudge' causes a wave, affecting the other object, which takes time. 2) there is how the already existing field affects the 'nudge'.

In the case where we nudge the magnet, 1) an electromagnetic wave travels from the magnet to the coil, induces a current, and a wave is sent back to the magnet, which causes a reaction force. (This is the delayed reaction, since it depends on whether the coil actually exists). and then there is 2) nudging the magnet means we are pushing a magnet through its own magnetic field, which causes a reaction on itself. This happens immediately, since the wave doesn't have to travel any distance.

In the case where we nudge the coil and the magnet is stationary: we have 1) the acceleration of charges in the coil through the stationary magnetic field causes an EM wave which travels from the coil to the magnet, which is affected, and sends a wave back to the coil, causing a reaction force on the coil. (Again this is the delayed reaction). And then there is 2) nudging the coil through the stationary magnetic field will cause a reaction on itself.

 

[entropy15]

There's two things going on 1) the 'nudge' causes a wave, affecting the other object, which takes time. In the case where we nudge the magnet

an electromagnetic wave travels from the magnet to the coil, induces a current, and a wave is sent back to the magnet

Are these waves real, like normal electromagnetic waves?
If the moving magnet is emitting an electromagnetic wave, it should do so only if the coil is present. Because if there was no coil there would be
nothing to absorb this wave. Also if there is no coil the magnet will not experience resistance meaning that it has not emitted the wave.
But how can the magnet know instantantaneously (as soon as it starts moving) that there is a coil at a distance x?

 

[BruceW]

Well, when the magnet is given a 'nudge', this will cause a change in the surrounding EM field which will propagate at c. (Which is due to the theory of retarded potentials in the Lorenz gauge). (And by 'propagate at c', I simply mean that after time t, the furthest point at which the EM field is affected by the nudge is at distance ct).

This is true whether or not the coil is there. And if the coil is there, it will take time x/c after the nudge, for the EM field at the coil to be affected. So it is at this time that the current can be induced in the coil, Then the propagation of this effect will again travel at c, so it will take another time of x/c for the magnet to experience the field which is due to currents which have been induced in the coil.

 

[BruceW]

I know I haven't immediately answered your questions, but I have not thought about this problem before, so I am trying to start with the principles that I am most certain should apply to this situation.

So, for the questions. Are they real waves? Um, I guess they satisfy the inhomogeneous wave equations. So yes? But then by this definition, every classical electromagnetic phenomena involves real waves. If we instead define a real wave as being oscillatory, then I guess generally the waves in this case would be partly real and partly not.

I would expect the accelerating magnet to emit EM energy even if the coil was not there. For example, if the magnet was made of a coil with current flowing through it, then when we nudge the magnet, we are accelerating charges, which generally gives off EM radiation.

 

[Drakkith]

Are these waves real, like normal electromagnetic waves?
If the moving magnet is emitting an electromagnetic wave, it should do so only if the coil is present. Because if there was no coil there would be
nothing to absorb this wave. Also if there is no coil the magnet will not experience resistance meaning that it has not emitted the wave.
But how can the magnet know instantantaneously (as soon as it starts moving) that there is a coil at a distance x?

I would guess that EM waves would be created by the acceleration, but I really don't know. I know accelerating a single charge will do this, but I don't know about a magnet. IF it emits EM waves then it must feel a small amount of "resistance" from the emission. This would be separate resistance from the resistance felt due to induction in the coil, and would only happen during the acceleration. Once it was moving steadily it would not emit EM radiation since it isn't accelerating. (That of course is IF it emits EM radiation in the first place)

 

[entropy15]

IF it emits EM waves then it must feel a small amount of "resistance" from the emission. This would be separate resistance from the resistance felt due to induction in the coil, and would only happen during the acceleration. Once it was moving steadily it would not emit EM radiation since it isn't accelerating. (That of course is IF it emits EM radiation in the first place)

The energy transfer here should be taking place due to the induction and not through radiation (which even if present should be small and can be neglected as well).

The delay for the magnet to experience resistance due to the current in the coil seems to be
2x/c.

Indeed, If we follow the classical approach the delay seems to be 2x/c. But if we look at it from QED perspective it seems to be 0. (Wheeler Feynman absorber theory)

Can we throw more light on it looking at it from the relativity perspective?
After all magnetic field can be explained as an effect due to the "length contraction" involving moving charges using theory of relativity.

I have seen papers which explain the origin of magnetic fields using relativity. Most of them have to do with the forces of attraction/repulsion between two wires carrying current.
But I have not come across any which explain electromagnetic induction with the same approach.

I know it would be too complicated, but in case any of you have come across it, please help.

 

[BruceW]

Drakkith has got a good point. There are two different situations 1) considering a magnet which is being given a 'nudge' by some person. 2) considering a magnet which is moving toward the coil, with no 'outsider' forces.

I have mostly been talking about what happens when the magnet is given a 'nudge' i.e. acceleration. In this case, there is a delay before the effect is felt by the coil. I am pretty sure that Wheeler Feynman absorber theory also predicts that there is a time delay. But because I don't know much QFT, I can't say with complete certainty.

Now in the case when the magnet is moving freely (not being 'nudged'), then I am not so sure about whether there is a delay in the reaction force being felt, because in this case, at any particular time, there is already a magnetic field traveling between the magnet and the coil (which is due to the magnet moving at an earlier time).

 

[Jano L.]

Are these waves real, like normal electromagnetic waves?
If the moving magnet is emitting an electromagnetic wave, it should do so only if the coil is present. Because if there was no coil there would be
nothing to absorb this wave. Also if there is no coil the magnet will not experience resistance meaning that it has not emitted the wave.
But how can the magnet know instantantaneously (as soon as it starts moving) that there is a coil at a distance x?

It is just normal electromagnetic wave. When the magnet is nudged, the surface currents on it get nudged too and electromagnetic waves start to propagate out of the surface. In standard theory with retarded fields, there is no immediate effect of the coil on the magnet; this comes only later, after time 2x/c.

However, the magnet will feel small resistance immediately. This is due to the fact that one part of the surface current will act on another, an in case the magnet is accelerated, these forces do not cancel each other but tend to act against the acceleration. This is sort of a "self-inductance" effect.

Later, after time 2x/c, the wave from the coil will come and damp the magnet as well. This will be typically much greater effect (sort of "mutual inductance").

 

[entropy15]

However, the magnet will feel small resistance immediately. This is due to the fact that one part of the surface current will act on another, an in case the magnet is accelerated, these forces do not cancel each other but tend to act against the acceleration. This is sort of a "self-inductance" effect.

Later, after time 2x/c, the wave from the coil will come and damp the magnet as well. This will be typically much greater effect (sort of "mutual inductance").

Since the self inductance effect is very small let's consider only the mutual inductance.

Suppose that the magnet is initially at rest.
It is given a push at time t=t1. Let's assume that the kinetic energy transferred to the magnet due to this push is E_k.
This kinetic energy may decrease immediately due to radiation or self inductance but the effect is negligible.

Now at time t=t1+(x/c) we have the current induced in the coil and it has gained energy due to this induced current.

Now since the magnet does not experience any resistance till t1+(2x/c) i.e any considerable resistance we can see that the magnet does not lose the kinetic energy acquired by it during the initial push till t1+(2x/c).

So between t1+(x/c) and t1+(2x/c) we have this energy in the coil as well as in the kinetic energy of the magnet.

The kinetic energy of the magnet is the cause for the current to be induced in the coil.
The magnet just transfers its kinetic energy to the coil through mutual induction.
So should not the kinetic energy of the magnet decrease before the current is induced in the coil?

 

[Jano L.]

The kinetic energy of the magnet is the cause for the current to be induced in the coil.

Not exactly. It is better to say that the electromagnetic forces due to the magnet drive the currents in the coil.

The magnet just transfers its kinetic energy to the coil through mutual induction.

This is also not accurate enough. If we want to use the energy concept, we can, but we have to keep in mind all contributions to the energy. In this case, besides the magnet and the coil, there is also energy distributed throughout the remaining space. It is better to say that when the wave hits the coil, the field in the vicinity of the coil and inside it supply the energy for the animation of the currents.

So should not the kinetic energy of the magnet decrease before the current is induced in the coil?

It should not. Again, try to think of this in terms of the force - the response from the coil has not arrived yet, so there is no force acting on the magnet in the time interval x/c,2x/c.

You are probably worried about conservation of energy, but I think there is no problem. The currents in the coil get their energy from the field near the wires. Eventually, the magnet will lose some kinetic energy, but this is merely an after-effect.

 

[entropy15]

You are probably worried about conservation of energy, but I think there is no problem. The currents in the coil get their energy from the field near the wires. Eventually, the magnet will lose some kinetic energy, but this is merely an after-effect.

The current strength is proportional to the velocity of the magnet. The faster the magnet moves, more is the current.
The energy stored inside the magnetic field is constant. If the current gets energy from the field surrounding the coil, how can it obtain more and more energy from this field as the velocity of the magnet increases.

If the magnet is moving with a sufficiently high velocity the current induced in the coil may increase beyond what the surrounding magnetic field can provide.

 

[entropy15]

It should not. Again, try to think of this in terms of the force - the response from the coil has not arrived yet, so there is no force acting on the magnet in the time interval x/c,2x/c.

If we consider the Wheeler Feynman absorber theory the response from the coil arrives as advanced waves from the future to the instant when the magnet starts to move.
So this should get rid of the energy conservation problems.

In this case the resistance is felt immediately rather than 2x/c.

 

[Jano L.]

If the current gets energy from the field surrounding the coil, how can it obtain more and more energy from this field as the velocity of the magnet increases.

The energy of the field is not just magnetic; it has electric component as well. This has to be so, for pure magnetic field could not induce currents in the coil.

The energy of the field is the higher the higher is the velocity of the magnet. We can be sure that there is always enough energy present to maintain the energy conservation, because we have the Poynting theorem; this shows that the energy is conserved locally, and moves through space like fluid..

I agree it is difficult to imagine this in such complicated process, but there is an alternative way of description, via EM forces, which makes this much more clear. According to the Faraday law, the magnitude of the electric field due to magnet animating the currents is the higher the higher is the velocity of the magnet; the higher the electric field, the stronger currents get induced.

 

[Dale]

The current strength is proportional to the velocity of the magnet. The faster the magnet moves, more is the current.
The energy stored inside the magnetic field is constant. If the current gets energy from the field surrounding the coil, how can it obtain more and more energy from this field as the velocity of the magnet increases.

When you have a moving magnetic field then you also induce an E field according to Faraday's law. This E field also contains energy. As you push the magnet faster your dB/dt is greater and therefore the induced E field is also greater. The extra energy you are worried about is in this induced E field.

I see the scenario as follows:
1) magnet and coil at rest wrt each other, no forces
2) force on magent to accelerate magnet (extra force required in order to generate E field)
3) EM wave propagates at c to coil
4) changing current induced in coil
5) changing field produced by coil
6) EM wave propagates at c to magnet
7) coil field opposes motion of magnet

I recommend against throwing unnecessary quantum concepts into any discussion which can be done purely classically. Generally it adds more confusion than understanding.

 

[BruceW]

Now since the magnet does not experience any resistance till t1+(2x/c) i.e any considerable resistance we can see that the magnet does not lose the kinetic energy acquired by it during the initial push till t1+(2x/c).

No, the point is that when we give the magnet a 'nudge', we are putting energy into the electromagnetic field. So there is an immediate resistance, but not due to mutual inductance. (We will feel a resistance even if there is no coil). And if there is a coil, then we also get a delayed resistance due to mutual inductance.

Edit: So I guess I'm saying that if we only considered mutual inductance, then energy is not conserved when we give the magnet a nudge. (But this is because we are not considering the whole picture).

 

[BruceW]

If the magnet is moving with a sufficiently high velocity the current induced in the coil may increase beyond what the surrounding magnetic field can provide.

I don't understand what you mean here.

 

[entropy15]

No, the point is that when we give the magnet a 'nudge', we are putting energy into the electromagnetic field. So there is an immediate resistance, but not due to mutual inductance. (We will feel a resistance even if there is no coil). And if there is a coil, then we also get a delayed resistance due to mutual inductance.

Hi Bruce,
Thanks for replying to my queries.

One more thing here, suppose that we replace the magnet with an electromagnet. The current in the electromagnet is controlled by a switch. Also we will remove the coil.(which was originally placed at a distance x)

Initially the electromagnet is not powered on and hence no magnetic field would be present.
Now we nudge this electromagnet so that it starts moving at a constant velocity.

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.

 

[entropy15]

I don't understand what you mean here.

I was assuming that the energy in the electric field was constant, but as DaleSpam has pointed out the faster the magnet moves more the energy in the electric field, so this should not be a problem.

 

[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?