Boundary conditions in Electrostatics

[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.

 

[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.

 

[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?

 

[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.

 

[sardar]

I would like to clarify that the boundary conditions in electrostatics are essential in determining the behavior of electric fields and potentials in different materials. The boundary conditions dictate the behavior of electric fields and potentials at the interface between different materials, such as a conductor and an insulator.

To answer your first question, there is indeed a difference in the boundary conditions when dealing with a grounded conducting material and an insulator around, compared to a non-charged insulated conducting material and an insulator around. In the case of a grounded conducting material, the potential inside the material will always be zero, as you correctly stated. However, in the case of a non-charged insulated conducting material, there can be a non-zero potential inside the material due to the presence of image charges. This is because the charges on the surface of the material can induce an electric field, which can result in a non-zero potential inside the material. Therefore, the boundary conditions are different in these two cases.

To answer your second question, it is possible to get a non-zero potential inside an uncharged insulated conducting material, as I mentioned earlier. This can happen due to the presence of image charges. However, the potential will be constant inside the material, as you correctly stated.

Regarding your third question, the substitution method described in Jackson's book is a useful technique for solving problems involving charged insulated conducting materials. It is based on the principle of superposition, where the total electric field and potential are the sum of the fields and potentials due to the individual charges present in the system. This method is not limited to just the example given in the book, and it can be applied to other situations as well. However, it is essential to note that this method is only valid for static problems. For dynamic problems, other techniques need to be used.

In conclusion, boundary conditions play a crucial role in determining the behavior of electric fields and potentials in different materials. It is essential to understand and apply the correct boundary conditions to accurately analyze and solve electrostatic problems.