How Do You Calculate Polarization Charge on a Dielectric-Covered Sphere?

[miew]

I have this problem for homework and I don't even know where to start... I would really appreciate any help !

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, outer radius b, and dielectric
constant κ .
(a) Calculate the surface polarization
charge density pol σ on the inside
and outside surfaces of the
dielectric (linear dielectric).
(b) What is the volume charge density
pol ρ of polarization charge inside
the dielectric? Is this surprising?
Recall:
σ pol = rP •nˆ and ρpol = −∇ •rP .

Homework Equations

I don't even know where to start...

The Attempt at a Solution

 

[anathema]

First, let's define some variables to make the solution easier to follow:

a = radius of the metal sphere
b = outer radius of the dielectric layer
Q = free charge on the surface of the metal sphere
κ = dielectric constant of the dielectric layer
σ = surface charge density
ρ = volume charge density
P = polarization vector
nˆ = unit normal vector
r = distance from the center of the sphere

(a) To calculate the surface polarization charge density, we need to find the polarization vector P. We can do this by using the formula P = κE, where E is the electric field inside the dielectric. Since the metal sphere is isolated, the electric field inside the dielectric will be due to the free charge Q on its surface. We can use Gauss's law to find the electric field at a distance r from the center of the sphere:

∮E•dA = Q/ε0

Since the electric field is radial and the surface area of a sphere is 4πr^2, we can rewrite this as:

E(4πr^2) = Q/ε0

Solving for E, we get:

E = Q/(4πr^2ε0)

Now, we can calculate the polarization vector P as:

P = κE = κQ/(4πr^2ε0)

To find the surface polarization charge density σ on the inside and outside surfaces of the dielectric, we can use the formula σ = rP•nˆ. Since the polarization vector is directed radially outward, we can rewrite this as:

σ = rP = rκQ/(4πr^2ε0)

For the inside surface of the dielectric (r = a), we get:

σinside = aκQ/(4πa^2ε0) = κQ/(4πaε0)

For the outside surface of the dielectric (r = b), we get:

σoutside = bκQ/(4πb^2ε0) = κQ/(4πbε0)

(b) To find the volume charge density of polarization charge inside the dielectric, we can use the formula ρ = -∇•P. Since the polarization vector is constant and directed radially outward, we can rewrite this as:

ρ = -∇•P = -∂P/∂r = -∂/