Describe the magnetic field anywhere around plate

[skrat]

Homework Statement

On a thin infinite metal plate, which is $d$ thick and has got electric conductivity $\sigma$, two electrodes are applied. The distance between the electrodes is $r$ and the potential is $U$.

Describe the magnetic field anywhere in space. Ignore the contribution of the lines that bring the current to electrodes.

Homework Equations

The Attempt at a Solution

I don't really know what would be the smartest way to start this problem.

Basically, if I understand the problem correctly, I can imagine the two electrodes as two electric charges and on that plane I should get currents like this http://www.phy.ntnu.edu.tw/ntnujava/snapshotejs/twopointcharges_2_20090117112750.gif . Now for a given point in 3D space I would have to somehow integrate over the entire metal plate. That's if I am not completely wrong.

Hmmm.. Why do you suggest? How should I start?

 

[ZetaOfThree]

Well, modeling the electrodes as point charges seems a little bit complicated... plus the potential difference between two point charges would be infinite. As a general rule of thumb when solving physics problems, try to make things as simple as possible. Did they suggest a way to model the electrodes at all?

 

[Orodruin]

The only thing relevant for the magnetic field is the currents flowing in the plate as a result of the applied potential. Some questions to get you started: What is the electric potential in the plate going to be? How does this relate to the current density? How does the current density relate to the magnetic field? And yes, somewhere there will be an integral over the metal plate.

 

[skrat]

Well, modeling the electrodes as point charges seems a little bit complicated... plus the potential difference between two point charges would be infinite. As a general rule of thumb when solving physics problems, try to make things as simple as possible. Did they suggest a way to model the electrodes at all?

Nono, they haven't. That was my interpretation of the problem which I though would simplify it, but according to you it does right the opposite.

The only thing relevant for the magnetic field is the currents flowing in the plate as a result of the applied potential.

I agree.

Some questions to get you started: What is the electric potential in the plate going to be? How does this relate to the current density? How does the current density relate to the magnetic field? And yes, somewhere there will be an integral over the metal plate.

What do you mean by "What is the electric potential going to be?" ? ?

 

[Orodruin]

From the problem, it is stated that the electrodes have a potential difference U. Now, the potential is only relevant up to a constant so we may use 0 and U (or +-U/2 for that matter, other choices seem annoying). It is this potential difference that drives the currents. If you prefer, you could work in terms of the electric field instead.

 

[skrat]

Aha, good point. Let's take +- U/2, just because it looks nice.

now to answer your other two questions;
-Hmmm, current density $j$ is now $j=\sigma E$, where $E$ is electric field. And I am tempted to say that $E=U/r$ but this doesn't seem right. I wouldn't expect the current density to be constant on the plane, therefore I think it should be $E=U/d$, where $d$ is (hmmm; this may be hard to explain with my knowledge of english) the length of the curve along which the current flows.
Or is it not?

-Relation between the current density and magnetic field is described with Amper's law.

 

[Oxvillian]

Homework Statement

On a thin infinite metal plate, which is $d$ thick and has got electric conductivity $\sigma$, two electrodes are applied. The distance between the electrodes is $r$ and the potential is $U$.

Describe the magnetic field anywhere in space. Ignore the contribution of the lines that bring the current to electrodes.

Is that the exact question?

Are the electrodes "infinitesimally small" or do they have some radius? Seems to me that the problem might not be mathematically well defined unless the electrodes have some nonzero size.

 

[skrat]

Is that the exact question?

Are the electrodes "infinitesimally small" or do they have some radius? Seems to me that the problem might not be mathematically well defined unless the electrodes have some nonzero size.

Yes, that is the exact question. But we were told that the problem has general parameters and if we wish to illustrate the solution or maybe even come up with a numerical result, we are allowed to determine meaningful parameters.

Basically, that is the only reason why I originally suggested modeling electrodes as point charges. Which turned out to be nonsense.

So my guess, and answer to you would be that we could probably limit ourselves on some shape of electrodes.

 

[Oxvillian]

Yes, that is the exact question. But we were told that the problem has general parameters and if we wish to illustrate the solution or maybe even come up with a numerical result, we are allowed to determine meaningful parameters.

Ah I see. That's probably good

Basically, that is the only reason why I originally suggested modeling electrodes as point charges. Which turned out to be nonsense.

I don't think it's complete nonsense, but if you're going to make the analogy work, you'll have to make it more exact.

What's $\nabla \cdot {\bf J}$?

 

[skrat]

$\nabla \cdot {\bf J}=-\frac{\partial \rho}{\partial t}$

 

[Oxvillian]

Which is what, given that the charge doesn't build up anywhere on the plate?

(except at the electrodes, maybe)

 

[skrat]

Which is what, given that the charge doesn't build up anywhere on the plate?

(except at the electrodes, maybe)

Well it has to be some kind of electric current. $I=\frac{de}{dt}=\frac U R$ where $R=\frac{1}{\sigma}\frac d S$ .

 

[Oxvillian]

Well, it's what the equation says - the rate of change of the charge density is minus the divergence of the current density

But what is the rate of change of charge density on the plate? Do we have blobs of charge in the middle of the plate getting bigger and bigger over time, or do we have a static situation?

 

[skrat]

:D

Static situation, of course.

 

[Oxvillian]

Great!

That means that $\frac{\partial \rho}{\partial t} = 0$. Which in turn means that $\nabla \cdot {\bf J} = 0$.

So what's $\nabla \cdot {\bf E}$ ?

(remembering that ${\bf J} = \sigma {\bf E}$)

 

[skrat]

I would say that $\nabla \cdot E=\frac{\rho }{\varepsilon _0}$ but since $\nabla \cdot {\bf J} = 0$, I would say it also has to be $0$.

 

[Oxvillian]

I agree.

So last question - what's $\nabla^2 \phi$, where $\phi$ is the potential?

(remembering that ${\bf E} = -\nabla \phi$)

 

[skrat]

It's $0$ , but i don't really get it what are you trying to say...

 

[Orodruin]

So $\nabla^2 \phi = 0$ in the plate. This means that the potential is a harmonic function and that in order to find the current you must solve Laplace equation. In order to do that you need some boundary conditions ...

 

[Orodruin]

I would say that $\nabla \cdot E=\frac{\rho }{\varepsilon _0}$ but since $\nabla \cdot {\bf J} = 0$, I would say it also has to be $0$.

By the way, this shows that if you have a conducting material, there are no internal residual charge densities in a stationary state.

 

[skrat]

So $\nabla^2 \phi = 0$ in the plate. This means that the potential is a harmonic function and that in order to find the current you must solve Laplace equation. In order to do that you need some boundary conditions ...

HA! It would take me years to figure that out! That's on the plate only, right?
huh, boundary conditions. Yeah, huh ... This can't be solved in polar coordinates right? Or is there anywhere a nice symmetry I don't see?
Ok, I guess, since we agreed that one electrode is at potential $U/2$ and the other is at $-U/2$, where the electrodes are $r$ apart, than we could set the origin of the coordinate system in the middle between the electrodes at $U=0$.


So the idea is to find the potential $\phi$, from there we have the current density as $J=-\sigma \nabla \phi$ and the final step would be i guess using the $\nabla \times B=\mu _0 J$

 

[Oxvillian]

It's $0$ , but i don't really get it what are you trying to say...

What we've basically done is reduced your physics problem to a math problem. We now know that on one electrode, the potential $\phi$ is $0$, that on the other electrode it's $U$ (or if you prefer $-U/2$ and $U/2$), and that between the electrodes, it obeys Laplace's equation $\nabla^2 \phi = 0$. As Orodruin says, it's now a standard "boundary value problem".

But what I'm really trying to say is that your original intuition about the analogy with the point charges turned out to be exactly correct!

That's because the potential $\phi$ in empty space due to an electrostatic charge distribution obeys that very same differential equation (Laplace's equation). Any solution you know about from electrostatics (eg. $\phi$ due to a point charge or a line charge or a sheet of charge) will automatically be a solution on your plate. But bear in mind that the plate is effectively 2-d, not 3-d...

[edit: sorry, posted this before reading your last reply]

 

[skrat]

Wowow, hold on.

That's only between the electrodes? To me that's a bit shocking now. :D

Where and why and how did we limit ourselves on the area between the electrodes? Especially why, since the problem is asking about the magnetic field anywhere in the 3D space, which reduces to a 2D problem if I understood correctly.

 

[Oxvillian]

Wowow, hold on.

That's only between the electrodes? To me that's a bit shocking now. :D

Probably I should say something like "anywhere on the plate that's not touching an electrode" instead of "between the electrodes".

Where and why and how did we limit ourselves on the area between the electrodes? Especially why, since the problem is asking about the magnetic field anywhere in the 3D space, which reduces to a 2D problem if I understood correctly.

The magnetic field is generated by the current in the plate, and the plate, being what they call a "thin" plate, is effectively two-dimensional.

To proceed any further, I think we have to go beyond what's given in the problem and decide on exactly what your electrodes look like. The electrodes are basically the boundary conditions for your boundary value problem.

 

[skrat]

This problem turned out to be way more complicated than I imagined. :D

I guess, if we take the electrodes as circles with radius $a<<r$ should be quite well. Don't you think so?

 

[Oxvillian]

That sounds sensible to me.

 

[skrat]

Btw, stupid question, what is the next step?
And even more stupid one: What is so special about the next step that we had to decide what the electrodes look like?

 

[Oxvillian]

skrat - I'm off on a road trip for a few days so maybe someone else can help...

[nudges Orodruin and ZetaOfThree ]

The electrodes are the boundary for your boundary value problem - without knowing where they are and what shape they are, you can't do anything.

Don't try to solve Laplace's equation directly - just pick an appropriate solution from electrostatics that fits, and that you already know about, and use that. In fact you already did in your very first post.

Good luck!

 

[skrat]

Let me know how the road trip was =))