Calculating Electric Fields in a Polarized Spherical Shell

[qspeechc]

Homework Statement

A thick, spherical shell (inner radius a, outer radius b) is made of dielectric material with a 'frozen in' polarization
$$\textbf{P} = \frac{k}{r} \textbf{\hat r}$$
where k is a constant and r the distance from the centre. (There is no free charge). Find the electric field in each o the three regions by two methods:

a) Locate all the bound charge, nd use Gauss' Law to calculate the field it produces
b) Use $$\oint \textbf{D}.d\textbf{a} = Q_{f enc}$$ to find $$\textbf{D}$$ and then find the electric field

The Attempt at a Solution

a) bound charge density is $$\rho _b -\nabla \textbf{P} = -frac{k}{r^2}$$

Gauss' Law: $$\oint E.da = Q_{enc}/\epsilon _0$$
In the dielectric material:
$$Q_{enc} = \int \rho _b = -4\pi \int _a^r dr '= 4\pi k(a-r)$$
I integrated from a to r because that's the only region with charge.
From Guass' Law:

$$E4\pi r^2 = 4\pi k(a-r)/\epsilon_0$$

$$E = k(a-r)/(\epsilon_0 r^2)$$

For r<a the electric field is zero because there is no bound charge. Similarly, outside the sphere
$$E = k(a-b)/(\epsilon_0 r^2)$$

b)
$$\oint \textbf{D}.d\textbf{a} = Q_{f enc} = 0$$
because there is no free charge. So
$$\textbf{D} = \epsilon _0 \textbf{E} + \textbf{P} = 0$$

$$\textbf{E} = -\textbf{P}/\epsilon_0$$

Where did I go wrong here?

 

[Jhero]

Your solution for the electric field using Gauss' Law is correct. However, in part b), you have made a mistake in your calculation for \textbf{D}. Since there is no free charge, the electric displacement \textbf{D} is equal to the polarization \textbf{P}. Therefore, your expression for \textbf{D} should be \textbf{D} = \textbf{P} = \frac{k}{r} \textbf{\hat r}.

Using this value for \textbf{D}, you can then use the relation \textbf{D} = \epsilon_0 \textbf{E} to find the electric field in each of the three regions. This will give you the same result as using Gauss' Law.

Overall, your approach is correct, but you just made a small mistake in your calculation for \textbf{D}. Keep up the good work!