Question

A metal sphere of radius a has a uniform (free) charge density σ_f on its surface. The permittivity of the dielectric region surrounding the sphere varies as , where r is the radial coordinate.

1. (1 pts) Determine the polarization P and electric field intensity E inside the sphere.
2. (3 pts) Determine the polarization P and electric field intensity E in the dielectric.
3. (5 pts) Calculate all bound charge densities, ρ_b and σ_{b. Is the dielectric homogeneous?}
4. (1 pts) Test whether the electric field intensity E in the dielectric is conservative.

Answer 1

The problem can be formulated as follows,

The charges on the spherical conductor comes up on the surface and distributes itself uniformly with charge density _f.

To calculatet the quantities demanded we will be applying Gauss's law.

D is the displacement vector and Q_f is the total free charge enclosed in the gaussian surface(S) considered.

We need some fundamental formulas like,

_e is the electric susceptibility.

Here it is stated in the question that the electric permittivity of the dielectric is depended on r only but the form is not given! So I am going to assume that is some function of r like,

.

Now, to get the electric field inside the conducting sphere we apply gauss's law for a gaussian surface inside sphere drawan at a distance r from the center (r <a)

As the charges all come up to the surface, there is no charges inside the conducting sphere therefore Q = 0. (Q is the total free charges inside the gaussian surface). Simply, the E inside the sphere will also 0.

Now for the E in dielectric, we draw a gaussian surface is the dielectric at a distace r from the center.

The total charge enclosed inside the surafce is given as,

The displacement vector may be calculated as,

According the gauss's theorem,

Now we have D we can get the electric field E and the polarization vector P as,

Note, they all point outwards in the radial direction.

We know that electric susceptibility can be written as,

putting this into expression P,

The bound surface charge density is now,

And the volume charge density,

NOTE: Untill I get the form of (r) i.e. how it depends on the radius we cannot be sure about the quantities.

Now to know if the dielectric is homogeneous or not, this boud charge density has to be 0.

Lets calculate the curl E of the dielectric,

The E inside the dielectric can be written as,

Now the curl E is,

Therefore the Electric field is conservative inside the dielctric.