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

• peguerosdc
In summary: The polarized sphere can be seen as a combination of a shell with surface charge ##\sigma_b## and a solid sphere with volume charge ##\rho_b##. This allows us to use the equations and theories we already know for those cases. However, we must be careful to take into account the non-uniformity of the surface bound charge density when calculating the electric field inside the sphere. In summary, Griffiths explains in section 4.2.1 that the field produced by a polarized material can be calculated by considering it as a combination of two objects: a shell with surface charge density ##\sigma_b## and a solid sphere with volume charge density ##\rho_b##. However, it is important to note
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?

peguerosdc said:
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.

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

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
peguerosdc said:
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.

## 1. What is the formula for the electric field of a uniformly polarized sphere?

The formula for the electric field of a uniformly polarized sphere can be expressed as E = (1/4πε₀) * (3P - rPcosθ)/r³, where P is the dipole moment of the sphere, r is the distance from the center of the sphere, and θ is the angle between the dipole moment and the position vector.

## 2. How does the electric field of a uniformly polarized sphere differ from that of a point charge?

The electric field of a uniformly polarized sphere is different from that of a point charge because it takes into account the dipole moment of the sphere, whereas a point charge only has a single charge at its center. This results in a more complex and non-uniform electric field around the sphere.

## 3. Can the electric field of a uniformly polarized sphere be affected by an external electric field?

Yes, the electric field of a uniformly polarized sphere can be affected by an external electric field. The external electric field can cause the dipole moments of the sphere to align in a different direction, resulting in a change in the overall electric field of the sphere.

## 4. How does the electric field of a uniformly polarized sphere change with distance from the center?

The electric field of a uniformly polarized sphere follows an inverse distance relationship, meaning that it decreases as the distance from the center of the sphere increases. This is similar to the electric field of a point charge.

## 5. Can the electric field of a uniformly polarized sphere be zero at any point?

Yes, the electric field of a uniformly polarized sphere can be zero at certain points. This occurs when the dipole moment of the sphere is aligned with the position vector, resulting in a cancellation of the electric field. These points are known as neutral points.

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