Electric field of infinite sheet with time dependent charge

[ramazkhomeriki]

Consider infinite charged sheet in xy plane and suppose that charge is gradually and uniformly removed from the sheet. Electric field outside the sheet obviously do not depend on x and y variables, thus the Maxwell equation divE=0 reduces to the simpler form ∂Ez/∂z=0, this means that z component of electric field is uniform along z at any moment. But then it follows that the change of charge of the sheet is felt simultaneously at any point along z. I have no idea where a mistake is.

 

[Dale]

Can charge change without a current?

 

[ramazkhomeriki]

Can charge change without a current?

Ok, let us suppose that there is a current as well, but the equation in free space divE =0 remains the same.

 

[Dale]

Ok, let us suppose that there is a current as well, but the equation in free space divE =0 remains the same.

Yes, but now you have a source, the current, that causes the change in E. And it does so, according to Jefimenko’s equation, in a way that preserves causality

 

[ramazkhomeriki]

Yes, but now you have a source, the current, that causes the change in E. And it does so, according to Jefimenko’s equation, in a way that preserves causality

Ok, I have a source, but from the symmetry reasoning still there is no dependence on x and y variables in free space (it is supposed that the charge is removed uniformly from any part of the sheet). thus, we still have the equation dEz/dz=0 and the homogeneous electric field along z at any time.

 

[phyzguy]

Where is the charge going? I think the current must be flowing away in the Z-direction It can't really flow away in the X and Y direction in a way that preserves X,Y symmetry. So if there is a charge flowing away in the Z-direction, then there is charge in the Z-direction away from the initial charge sheet, no?

 

[ramazkhomeriki]

Where is the charge going? I think the current must be flowing away in the Z-direction It can't really flow away in the X and Y direction in a way that preserves X,Y symmetry. So if there is a charge flowing away in the Z-direction, then there is charge in the Z-direction away from the initial charge sheet, no?

Yes, sure... Then it follows that z component of electric field does not change in time?

 

[Dale]

but from the symmetry reasoning still there is no dependence on x and y variables in free space

Yes, that is correct and desired, by symmetry.

we still have the equation dEz/dz=0 and the homogeneous electric field along z at any time.

That no longer holds except in the special case where the current has been constant forever. In that special case the homogeneity in z is correct.

If the current varies over time, however, then the field is no longer homogenous in z and the changes in the z direction propagate causally

Look at all three terms in Jefimenko’s equation for the E field.

 

[vanhees71]

Consider infinite charged sheet in xy plane and suppose that charge is gradually and uniformly removed from the sheet. Electric field outside the sheet obviously do not depend on x and y variables, thus the Maxwell equation divE=0 reduces to the simpler form ∂Ez/∂z=0, this means that z component of electric field is uniform along z at any moment. But then it follows that the change of charge of the sheet is felt simultaneously at any point along z. I have no idea where a mistake is.

But $\vec{\nabla} \cdot \vec{E}=0$ is a contradiction to your setup, because you say you have a surface charge on the sheet. The correct equation is (in SI units)
$$\vec{\nabla} \cdot \vec{E}=\rho/\epsilon_0=\delta(z) \sigma(t,x,y)/\epsilon_0.$$
Of course everything else said in this thread so far is correct too, i.e., if $\partial_t \sigma$ you also must have a current density such that the continuity equation, $\partial_t \rho+\vec{\nabla} \cdot \vec{j}=0$ is fulfilled. Given both the charge and current density you get the fields via the retarded potentials (or, usually more cumbersome directly the em. field using the corresponding "Jefimenko equations").

 

[Vanadium 50]

If the current varies over time

It must, because eventually you will run out of charge.

If I were asked to solve this, I would start by modeling the charge density as exponentially decaying with time. Physically plausible, and I have a hope of getting a solvable differential equation.

 

[phyzguy]

Yes, sure... Then it follows that z component of electric field does not change in time?

How could that possibly be true? As @Vanadium 50 pointed out, the amount of charge is decreasing, so the current has to be decreasing, so the electric field must be a function of time.

 

[Vanadium 50]

I think the OP is getting tangled up in knots by making mutually contradictory assumptions - a static system, instantaneous reaction time, etc.

Consider you have an infinite sheet, open on top, and attached to a resistive medium on the bottom, and grounded underneath that. That's an RC circuit - your capacitor will discharge into ground. As this happens, the field above it will decrease.

Normally, we consider the approximation that everything happens slowly compared to the light travel time between the elements: true for most typical RC time constants and the size of capacitors. If that approximation isn't good enough, you have to roll up your sleeves and use retarted potentials or similar.

 

[ramazkhomeriki]

I think the OP is getting tangled up in knots by making mutually contradictory assumptions - a static system, instantaneous reaction time, etc.

Consider you have an infinite sheet, open on top, and attached to a resistive medium on the bottom, and grounded underneath that. That's an RC circuit - your capacitor will discharge into ground. As this happens, the field above it will decrease.

Normally, we consider the approximation that everything happens slowly compared to the light travel time between the elements: true for most typical RC time constants and the size of capacitors. If that approximation isn't good enough, you have to roll up your sleeves and use retarted potentials or similar.

The entire problem is to have two sheets of charges (one positive and one negative), i.e. capacitor and then one shortens those two sheets. Of course, some electromagnetic waves will propagate inside the capacitor, but those waves are transverse, not changing z component of electric field. My problem is to calculate how E_z changes in time by simplest argument that the problem is homogeneous in xy plane, but it seems that it is not possible. I thought that this problem should be simpler than the one with point like time dependent charge...

 

[phyzguy]

The entire problem is to have two sheets of charges (one positive and one negative), i.e. capacitor and then one shortens those two sheets. Of course, some electromagnetic waves will propagate inside the capacitor, but those waves are transverse, not changing z component of electric field. My problem is to calculate how E_z changes in time by simplest argument that the problem is homogeneous in xy plane, but it seems that it is not possible. I thought that this problem should be simpler than the one with point like time dependent charge...

Can you explain further? If the two sheets extend in the X and Y directions, what do you mean by "one shortens those two sheets"? Do you mean they move closer together in Z? Or do you mean their X,Y extent is reduced?

 

[ramazkhomeriki]

Can you explain further? If the two sheets extend in the X and Y directions, what do you mean by "one shortens those two sheets"? Do you mean they move closer together in Z? Or do you mean their X,Y extent is reduced?

I mean just to connect those two sheets by wire.

 

[phyzguy]

OK. So then what's the question? As @Vanadium 50 said, "Normally, we consider the approximation that everything happens slowly compared to the light travel time between the elements: true for most typical RC time constants and the size of capacitors."

In that approximation, Ez just decays exponentially.
If you're not satisfied with that approximation, then the problem is much more complicated. Electromagnetic waves will propagate laterally from the connection point of the wire, and vertically between the plates.

 

[ramazkhomeriki]

OK. So then what's the question? As @Vanadium 50 said, "Normally, we consider the approximation that everything happens slowly compared to the light travel time between the elements: true for most typical RC time constants and the size of capacitors."

In that approximation, Ez just decays exponentially.
If you're not satisfied with that approximation, then the problem is much more complicated. Electromagnetic waves will propagate laterally from the connection point of the wire, and vertically between the plates.

You say, Ez varies much slower than E_y and E_x, because the change of E_z is caused by slow movement of charges, while E_y and E_z are changing due to wave propagation. Do I understand correctly? It would simplify the problem but one has to be sure that the propagation of electromagnetic waves will not alter E_z.

 

[phyzguy]

You say, Ez varies much slower than E_y and E_x, because the change of E_z is caused by slow movement of charges, while E_y and E_z are changing due to wave propagation.

Where did I say that??

 

[ramazkhomeriki]

Where did I say that??

You said: "Ez just decays exponentially". I guess decay rate is just a time is needed for the charge to travel in the wire from one sheet to another.

 

[Vanadium 50]

Let's take a strep back. You were given a setup with an approximate answer. Is that not good enough? After all - the setup is either an approximation (infinite f`or really big) or unphysical (infinite is really infinite).

You've also been given the exact solutions, if by name and not by equation (twice - once in terms of fields and once in terms of potentials) Why isn't this good enough? "Here's a tough problem - you guys solve it" tends not to generate many (often any) takers.

What do you want that hasn't already been said?

 

[ramazkhomeriki]

Let's take a strep back. You were given a setup with an approximate answer. Is that not good enough? After all - the setup is either an approximation (infinite f`or really big) or unphysical (infinite is really infinite).

You've also been given the exact solutions, if by name and not by equation (twice - once in terms of fields and once in terms of potentials) Why isn't this good enough? "Here's a tough problem - you guys solve it" tends not to generate many (often any) takers.

What do you want that hasn't already been said?

In reality I have much more complex physical system to solve, that is multiferroic BiFeO3 domain wall structure (here is the information: https://pubs.acs.org/doi/full/10.1021/acs.nanolett.9b03484). There were some activities on generating THz radiation charging and then shortening ultimate sides of that structure. I am trying to simplify the picture by introducing capacitor system coupled in series, I thought that the problem I have formulated was well known and easily solvable, as I see it is not the case. Thank you for your answers...

 

[Ibix]

shortening

Just for clarity, "shortening" something means to reduce its length, while "shorting" something means to make an electrical connection with zero resistance. I think you mean the latter.

 

[ramazkhomeriki]

Just for clarity, "shortening" something means to reduce its length, while "shorting" something means to make an electrical connection with zero resistance. I think you mean the latter.

Yes, I meant that...