Conducting Sphere Covered By Spherical Dielectric

[csnsc14320]

Homework Statement

An isolated metal sphere of radius a has a free charge Q on its surface. The sphere is covered with a dielectric layer with inner radius a and outer radius b

Calculate the polarization charge density on the inside and outside of the dielectric.

Homework Equations

The Attempt at a Solution

So I know that the electric field outside of the conducting sphere w/o dielectric is:

$$E = \frac{k Q}{r^2}$$

This field however, should have the electric field due to the dielectric subtracted from it.

If we can regard the dielectric as a capacitor since it has an equal amount of polarized charge on its inner and outer surfaces, the electric field for a < r < b should be:

$$E = \frac{k q'}{r^2}$$ where q'=charge on inner surface of dielectric

the charge on the inner surface should just be

$$q' = \sigma_{inner} 4 \pi a^2$$


now I know I want $$\sigma_{inner}$$, and then it would be simple to find $$\sigma_{outer}$$, but I am not really sure how to solve for it

 

[csnsc14320]

or is this totally wrong?

 

[kerimek]

.



I would approach this problem by first understanding the concept of polarization charge density. This is the charge that is induced on the surface of a dielectric material when it is placed in an electric field. In this case, we have a conducting sphere covered with a dielectric, so we need to consider the polarization charge on both the inside and outside surfaces of the dielectric.

To find the polarization charge density on the inside surface of the dielectric, we can use the equation q' = \sigma_{inner} 4 \pi a^2, as mentioned in the given attempt at a solution. This equation relates the surface charge density, \sigma_{inner}, to the total induced charge, q', on the inner surface of the dielectric. We can rearrange this equation to solve for \sigma_{inner}, which would give us the polarization charge density on the inside surface of the dielectric.

To find the polarization charge density on the outside surface of the dielectric, we can use the same approach. The only difference is that the total induced charge on the outside surface, q', will be equal to the total free charge on the conducting sphere, Q, since the dielectric layer is covering the sphere. We can then solve for \sigma_{outer} using the same equation as before.

It is important to note that the electric field inside the dielectric will be different from the electric field outside the dielectric. This is because the presence of the dielectric material will affect the electric field, as mentioned in the given attempt at a solution. Therefore, we need to consider this when calculating the polarization charge density on both surfaces of the dielectric.

In conclusion, to find the polarization charge density on the inside and outside surfaces of the dielectric, we can use the equations q' = \sigma_{inner} 4 \pi a^2 and q' = Q, respectively, and solve for \sigma_{inner} and \sigma_{outer}. This will give us the required information to understand the behavior of the electric field in this system.