Why can we assume dielectric induced dipoles are pure dipoles?

[Happiness]

Homework Statement

Griffiths was trying to prove that when calculating the electric field inside a dielectric, we may assume the dipoles induced in it are "pure" dipoles, although they are in fact "physical" dipoles, as long as we view the field as a macroscopic field, one that is averaged over a sufficiently large region of space.

I don't follow his argument in the last paragraph of page 174, the one just before equation (4.19).

Homework Equations

Page 174 (attached) of "Introduction to electrodynamics" by Griffiths

The Attempt at a Solution

I calculated that

$V(r)=\frac{1}{4\pi\epsilon_0}\int_{inside}\frac{\hat{\eta}\cdot P(r')}{\eta^2}d\tau'=0$

if $P(r')$ is constant throughout the inside volume. ($\eta$ is Griffiths' script r.)

How does this relate to equation (4.19) and the argument about using a uniformly polarised sphere?

Pages 173-175 (Enlarged)

The same pages in their original size:

 

[mark726]

Thank you for bringing up your question regarding Griffiths' argument on the calculation of electric fields inside a dielectric. After reviewing the pages in question, I believe I can provide some clarification.

In the last paragraph of page 174, Griffiths is discussing the case of a uniformly polarized sphere, where the polarization vector P is constant throughout the entire sphere. In this case, the electric field inside the sphere can be calculated using the equation (4.19) that you mentioned. This equation takes into account the fact that the polarization vector is constant and relates it to the electric field.

In your attempt at a solution, you have correctly calculated that the potential inside the sphere is zero if the polarization vector is constant. This is in line with Griffiths' argument that we can view the induced dipoles as "pure" dipoles, as long as we are considering the macroscopic field averaged over a sufficiently large region.

To further understand this argument, it may be helpful to consider the fact that in a dielectric material, the individual molecules may have randomly oriented dipoles. However, when we consider the material as a whole, the average dipole moment is in a particular direction, giving rise to a macroscopic polarization vector. This is the concept of "polarization" in a dielectric material.

I hope this helps to clarify the argument and its relation to equation (4.19). If you have any further questions, please do not hesitate to ask.