Calculating Electric Field of a Polarized Sphere

[abode_x]

From Griffiths, Problem 4.1
A sphere of radius R carries a polarization
$$\textbf{P}=k\textbf{r}$$

where k is constant and r is the vector from the center.
a. Calculate $$\sigma_b$$ and $$\rho_b$$.
b. Find the field inside and outside the sphere.

part a is handled simply by $$\sigma_b=\textbf{P}\cdot\hat{n}$$ and $$\rho_b=-\nabla\cdot\textbf{P}$$.

part b is handled most easily by using the bound charges found and gauss's law, giving: $$\textbf{E}=\frac{-kr}{\epsilon_0}\hat{r}$$ and 0 outside.

part b can also be handled by first getting the potential through:$$\frac{1}{4\pi \epsilon_0}\int_v \frac{\hat{r}\cdot\textbf{P}}{r^2}d\tau$$. Then just get the negative gradient of V to yield E. This gives the same result as method 1 for both inside and outside (P=0 when dealing with outside case).

Now, i tried to get the field, again by computing for V but this time with:
$$V(r)=\frac{1}{4\pi \epsilon_0}\oint_s\frac{\sigma_b}{r}da + \frac{1}{4\pi \epsilon_0}\int_v\frac{\rho_b}{r}d\tau$$. The answer i get using this integral is different. i would like to know if in the first place this integral is applicable? if I'm trying to get the potential inside, do i still consider the surface bound charge?

the best i managed was an answer negative to the correct one. (but this had 1 dubioius step involved). also, i could only attempt to get the potential inside. can anyone make the last method work (for both outside and inside)? or share why it doesnt?

thanks.

 

[Meir Achuz]

\sigma_b is not spherically symmetric, so E iinside is not.
You can't use Gauss as you do in that case.
Your integrals should give the right answer if you mean r-r'
when you write r.

 

[abode_x]

Gauss's law actually gives the right answers. $$\sigma_b=kR$$ everywhere on the surface if you apply the definition. likewise, $$\rho_b=-3k$$. these are symmetric and so is E inside and out.

as for the r-r' hint. i don't see it yet, but i will try it. thanks.

i was wrong though, about my claim that i got E=0 outside using the second method. that's another issue - how to get that result.

 

[Meir Achuz]

Gauss's law actually gives the right answers. $$\sigma_b=kR$$ everywhere on the surface if you apply the definition. likewise, $$\rho_b=-3k$$. these are symmetric and so is E inside and out.

as for the r-r' hint. i don't see it yet, but i will try it. thanks.

i was wrong though, about my claim that i got E=0 outside using the second method. that's another issue - how to get that result.

\sigma_b does not equal kR. It equals kR cos\theta because of the dot product. Go back and study Griffriths some more, then come back to the problem.

 

[siddharth]

\sigma_b does not equal kR. It equals kR cos\theta because of the dot product. Go back and study Griffriths some more, then come back to the problem.

Meir Achuz, I don't quite understand this.

If $P(r)=kr \vec{e_r}$
Then, won't $$\sigma_b$$ be

$$\sigma_b = \vec{P}.\hat{n} = k R (\vec{e_r}.\vec{e_r}) = kR$$

 

[Meir Achuz]

Whoops. I apologize. I had read it as P=a fixed vector.
Using Gauss as in the first post gives the right answer.
To use the potential integrals still requires r-r'.
Thanks for catching my carelessness.