[Griffiths ex4.2] Electric field of a uniformly polarized sphere

[peguerosdc]

Homework Statement

Find the electric field produced by a uniformly polarized sphere of radius R

Relevant Equations

$\sigma_b = \boldsymbol{P} \cdot \boldsymbol{\hat{n}}$
$\rho_b = - \nabla \cdot \boldsymbol{P}$

Hi!

This is more a conceptual question rather than the calculation itself.

So, Griffiths' section 4.2.1 "The field of a polarized object / Bound charges" says that if you want to calculate the field produced by a polarized material, you can find it from the potential of a surface charge and a volume charge like this:

$V(\mathbf{r}) = \frac {1} {4\pi\epsilon_0} \oint \frac {\sigma_b} {r - r'} da' + \frac {1} {4\pi\epsilon_0} \int \frac {\rho_b} {r - r'} d\tau'$

which are called "bound charges".

As I understand this, it means that you can see the polarized object as a combination of two objects i.e. for the case of a polarized sphere of radius R, you can consider it as a shell of radius R with a surface charge density $\sigma_b$ and a solid sphere of radius R with volume charge density $\rho_b$ and use all the theory we already know for those cases i.e. Gauss's law (if useful), definition of E, etc.

Is this correct? The reason I ask (and the reason I suspect it is not) is because in example 4.2, Griffiths asks you to find the electric field produced by a uniformly polarized sphere of radius R, but as the sphere is uniformly polarized, then $\rho_b=0$, so it should mean that I can consider the polarized sphere as only a spherical shell with charge density $\sigma_b$, right?. If my assumption is correct, then the electric field inside the polarized sphere should be 0 (as it would be the field only due to the spherical shell) but NO! Griffiths says there is a non-constant potential inside the polarized sphere and, thus, a non-zero electric field inside.

Then, why is my assumption of making a polarized object a combination of surface + volume objects not correct if, apparently, from the equation of V it should?

 

[PeroK]

The reason I ask (and the reason I suspect it is not) is because in example 4.2, Griffiths asks you to find the electric field produced by a uniformly polarized sphere of radius R, but as the sphere is uniformly polarized, then $\rho_b=0$, so it should mean that I can consider the polarized sphere as only a spherical shell with charge density $\sigma_b$, right?. If my assumption is correct, then the electric field inside the polarized sphere should be 0 (as it would be the field only due to the spherical shell) but NO! Griffiths says there is a non-constant potential inside the polarized sphere and, thus, a non-zero electric field inside.

Then, why is my assumption of making a polarized object a combination of surface + volume objects not correct if, apparently, from the equation of V it should?

The surface charge density is not uniform, so the field inside is not zero.

 

[robphy]

Following what @PeroK says, the surface bound charge density $\sigma_b \neq \overrightarrow{(constant)}$.
Why is this so? What is your definition of $\sigma_b$?

 

[peguerosdc]

The definition is $\sigma_b = \vec P \cdot \hat n = P \cos(\theta)$ so, right! It is not uniform so the field is not zero inside the sphere.

Such an obvious error, but lesson learned: if P is uniform, that doesn´'t mean the densities are going to be uniform. Thanks!

Besides from that, just to confirm, is my assumption that we can now consider the problem as a combination of surface + volume objects correct?

 

[PeroK]

Besides from that, just to confirm, is my assumption that we can now consider the problem as a combination of surface + volume objects correct?

Yes, that's the whole point.