# Boundary conditions in Electrostatics

1. May 14, 2014

### Gavroy

If I have a grounded conducting material, then I know that $\phi=0$ inside this material, no matter what the electric configuration in the surrounding will be.

Now I have a conducting material that is not grounded, then there will be (as long as I am dealing with static problems) no electric field inside this material. Therefore the potential will be constant inside this material, right?

Question 1:Therefore, is there any difference in the boundary conditions if I am dealing with a grounded conducting material and an insulator around or a non-charged insulated conducting material and an insulator around?

Question 2:Is it possible to get a non-zero potential inside an uncharged insulated conducting material? Especially, would you get a non-zero potential inside a conducting insulated material due to image charges?

Question 3:Of course, I read a few pages in Jackson's book about this and saw that he substituted the problem of a charged insulated conducting sphere in an external field with the one of having a grounded conducting sphere in the external field that has a charge sitting in the center of the sphere. Then, the magnitude of the extra charge was given by the difference of the initial charge of the sphere minus the induced image charge on the grounded conducting sphere.

Is it possible to make a general substitution like this: Thereby I mean, that we substitute a charged insulated conducting material carrying a charge by a grounded conducting material that has an additional charge(magnitude given by the difference of total charge-image charge) sitting on its surface? So, I would solve the grounded problem and would add the difference of the total charge-image charge to the surface of the material and add this field to the field calculated for the problem of the grounded material.

2. May 14, 2014

### Simon Bridge

Q1. Yes - i.e. the insulator gives two extra boundaries to consider.
Less trivially, grounded conductors always have neutral charge everywhere while insulated conductors may become polarized in the presence of an external field. Does that not suggest that you have to treat the calculations differently?

Q2. Yes. you can apply a PD across a conductor - but it will no longer be a static system.
You can find out if the constant potential inside a static conductor can be non-zero by investigation using gauss' law.

Q3. You can make any substitution you like as long as the math works out.
Jackson's approach looks OK for other problems to me - check the situation though, it may only work for the specific geometry used.

Note: when the situation changes, you may need to change the model too.
That grounded sphere with a central charge will probably not behave exactly like the original insulated sphere all the time.

3. May 15, 2014

### Gavroy

1.)Why can a grounded conductor not become polarized?
2.) so how does Gauß help, it only tolls me that there is no e field.
3.) so you are basically saying: Maybe yes, Maybe no?

4. May 15, 2014

### Simon Bridge

1. depends on the field - how do you charge by induction?
That should show you how a grounded conductor works differently to an isolated one.

2. in the differential form - you may see it better with Poisson's equation.

3. yes.