E-field between insulator and conductor

[ross]

Homework Statement

I have a slab of conductor on the ground with thickness, a, and it extends in all directions infinitely. Then I have a slab of insulator above the conductor of thickness, a, and it also extends infinitely in all directions. The two slabs are a distance, 2a, from each other. The conductor is grounded. The insulator has some charge density p. It induces a charge density of -pa on the top surface of the conductor. The bottom surface of the conductor has zero charge density. The question assumes the E-field is always in the z-direction, and asks for the E-field for all values of z. Note that the situation is in electrostatic equilibrium.

Homework Equations

none really..

The Attempt at a Solution

I decided to split up the question into two parts: find the E-field due to the conductor for all z values, assuming the insulator isn't there.. then find the E-field due to the insulator, assuming the conductor isn't there.. then by the principle of super-position, add them up, and have the answer.

For conductors I know that eE = D, where e is epsilon, E is the E-field strength, and D is the surface charge density. So above the conductor, the E-field is E = (-pa)/e pointing up. The E-field within and below the conductor is zero.

For the insulator, I found the E-field above it to be E = (pa)/(2e) pointing up, below the insulator to be E = (-pa)/(2e), and within the insulator to be E = (p(z - 3.5a))/(e) pointing up.

My main question is in regards to below the conductor, because when I add up the E-fields generated by both the insulator and conductor, I get an E-field below the conductor, which I know is not possible. Is there some concept I'm missing here? Does the principle of superposition not work here? Maybe it's because the conductor is grounded? If possible, could you start me down the correct path for sovling this..? or just correct where I went wrong?

Thanks a bunch.

 

[ross]

Anyone know?

 

[Moetasim]

Hello,

Thank you for your question. It seems like you have a good understanding of the basics of electrostatics and the concept of superposition. However, there are a few things that need to be clarified in order to fully answer your question.

Firstly, the concept of superposition applies to linear systems, which means that the total effect is the sum of the individual effects. In your case, the system is not linear because the conductor is grounded. This means that the induced charge on the conductor's surface is not solely dependent on the charge density of the insulator, but also on the potential difference between the conductor and ground. Therefore, the principle of superposition cannot be applied in this case.

Secondly, the E-field below the conductor is not zero because there is a potential difference between the conductor and ground. This potential difference leads to a non-zero E-field, which decreases as you move further away from the conductor.

To properly solve this problem, you would need to use the concept of boundary conditions at the interface between the conductor and insulator. This means that the E-field must be continuous at the interface, and the tangential component of the electric displacement (D) must also be continuous. Using these conditions, you can find the E-field for all values of z.

I hope this helps guide you in the right direction for solving this problem. It's great to see you thinking critically and breaking the problem down into smaller parts. Keep up the good work!