Conductor thought experiment: Speed of E field

[FallenApple]

I wonder if my thought experiment in proving the finite speed of field lines is valid.

Say a spherical conductor has a cavity with positive charge Q inside as shown.
say that the beginning of the universe starts at time 0, where there is the charge Q in the cavity and that the conductor is not polarized. We are speaking in a classical perspective, so it makes sense.

Also, assume that the charge Q and the conductor are the only things in the entire universe.

In the time at the beginning of the universe, there is a field due Q outwards, existing everywhere in the universe at once simultaneously, and no field at all due to the conductor. Then afterwards Q does work to polarize the conductor.If we consider the system as the cavity’s Q and the conductor, then before the induction, the total field energy in the universe is soley to Q. After that brief moment, the total field energy of the universe should still the same ,since the electric force is a conservative force.

But we know that the energy is not conserved since the field lines are now terminated inside the conductor, leaving lesser length of field lines total in universe. Hence, less energy compared to time 0, since there was no gap.

This concludes that the original assumption that the field lines are existing simulataneously every where at once at the inception of time is wrong.

Therefore E has a finite speed.

So at the beginning of time, as E field from the inner cavity charge moves outwards to the conductor, it does work on the conductor as the E field itself is propagating at the same time. Hence there would be no contradiction. As the E field flows into the conductor, it does positive work on the negative charges in the conductor and positive work to the positive charges. So energy was put into creating the speration and therefore, the loss of the red field lines(inside the picture) was the price to pay. And hence, energy is conserved.

 

[Dale]

And hence, energy is conserved

Except that energy is not conserved in the field as you have described it. The energy density of the field is proportional to the square of the field. If the field just naively propagates to fill an increasing volume at the same energy density then it contains an ever increasing amount of energy without any input.

Instead of trying to invent a propagation speed for the electric field you should study the known propagation speed of the electromagnetic field.

 

[Dale]

If we consider the system as the cavity’s Q and the conductor, then before the induction, the total field energy in the universe is soley to Q. After that brief moment, the total field energy of the universe should still the same ,since the electric force is a conservative force.

But we know that is not true since the field lines are now terminated inside the conductor.

This section is also not correct. There is nothing wrong with field lines terminating on the inside of the conductor. Also, the field energy does not need to remain constant, provided the change in the field energy is equal to the work done on matter. This is what Poynting theorem describes.

 

[FallenApple]

Except that energy is not conserved in the field as you have described it. The energy density of the field is proportional to the square of the field. If the field just naively propagates to fill an increasing volume at the same energy density then it contains an ever increasing amount of energy without any input.

Instead of trying to invent a propagation speed for the electric field you should study the known propagation speed of the electromagnetic field.

I didn't assume that the energy would keep increasing. I know that the density falls off as 1 over distance squared.

So is it not conserved because while it does work, the charges on the conductor eventually stop moving? Kind of like how when gravitational potential energy gets lost due to friction?

 

[Dale]

So is it not conserved because while it does work, the charges on the conductor eventually stop moving? Kind of like how when gravitational potential energy gets lost due to friction?

No, the conductor is irrelevant to the issue about the field energy increasing over time. As you described it the energy in the field itself continually increases (before it even propagates out to the conductor).

Go ahead and work out the math. Write down the field as a function of t, then square that to get the energy density and integrate over all space to get the total energy. Is it constant?

 

[FallenApple]

This section is also not correct. There is nothing wrong with field lines terminating on the inside of the conductor. Also, the field energy does not need to remain constant, provided the change in the field energy is equal to the work done on matter. This is what Poynting theorem describes.

If we assume that the field lines are infinite everywhere, the before the conductor got polarized, the field had a certain energy inside the conductor. And the only energy in the universe is due to the single charge Q.

After the conductor got polarized, that original energy inside the conductor is gone. Where did it go?

Does the conductor have higher potential then before, due to the work?

 

[Dale]

After the conductor got polarized, that original energy inside the conductor is gone. Where did it go?

What does Poynting's theorem say about it?

This part is a very minor issue compared to the field energy issue, but it is easy to solve with Poynting's theorem.

 

[FallenApple]

What does Poynting's theorem say about it?

This part is a very minor issue compared to the field energy issue, but it is easy to solve with Poynting's theorem.

"the time rate of change of electromagnetic energy within V plus
the net energy flowing out of V through S per unit time is equal
to the negative of the total work done on the charges within V"

That is the definition. So does that mean that the energy that flowed out is replaced by the energy that the conductor gained?

That makes sense, even though the conductor has no voltage, the potential is different from before due to work.

 

[FallenApple]

No, the conductor is irrelevant to the issue about the field energy increasing over time. As you described it the energy in the field itself continually increases (before it even propagates out to the conductor).

Go ahead and work out the math. Write down the field as a function of t, then square that to get the energy density and integrate over all space to get the total energy. Is it constant?

I'm not at that point in the textbook that deals of time varying fields. And I rather have a feel for it, which mathematical derivations often don't provide.

But I suppose intuitively it makes sense. Energy can't be created out of nowhere. So the energy do to a point charge over all of space at a particular time must be constant. Therefore while the field lines are getting longer, it needs to cover larger area and they are getting weaker at the same time

Also, the electric field only acts when a test charge is there.

 

[Dale]

I'm not at that point in the textbook that deals of time varying fields.

You should wait until then. Your thought experiment is highly problematic, and the correct treatment requires knowledge of the time varying fields.