- #1

peguerosdc

- 28

- 7

- 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?

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?