How Does Polarization Affect Electric Field Calculation in a Dielectric Disc?

[Saketh]

I'm trying to teach myself polarization and dielectrics by doing problems, but it's not going as well as I'd hoped. Here's the first problem that I got stuck on:

A round dielectric disc of radius R is statically polarized so that it gains the uniform polarization P, with the vector P lying in the plane of the disc. Find the strength E of the electric field at the center of the disc if d << R.​

I thought that since the disc is polarized along its plane, there would be a charge density $$\sigma$$ on one half of it, and a charge density of $$-\sigma$$ on the other half of it. I wasn't sure if $$\sigma = P$$, but I set them equal anyway. I also wasn't sure if $$\sigma$$ is uniform along each half-surface, but I did that anyway. Then I thought that a uniform polarization vector means that the whole surface is charged up unformly, which means no net electric field at the center.

I realize this is going to sound silly, but I have no idea how I'm supposed to solve this problem. Where am I supposed to start?

 

[member 553083]

The first thing you need to do is to determine the charge distribution on the dielectric disc. Since the polarization is uniform, this means that the same amount of charge will be distributed uniformly across the surface of the disc. To calculate this, you can use the equation \sigma = \epsilon_0 E P, where \epsilon_0 is the permittivity of free space and E is the electric field at the surface of the disc. Once you have this, you can then calculate the electric field at the center of the disc by using the equation E = \frac{1}{4 \pi \epsilon_0}\frac{\sigma}{r^2}, where r is the distance from the center of the disc to the point where the electric field is being measured.