Field inside spherical hole inside dielectric

[sergiokapone]

Let we have a dielectric with field $E$ inside and with a little hole. I have problem. I get a two different answers on this problem, and I try to understand which one of them correct.

As mentioned in http://www.feynmanlectures.caltech.edu/II_11.html#Ch11-S4 (11.25), the electric field in cavity (in SGS)
\begin{equation}
E_{hole} = E + \frac{4\pi}{3}P
\end{equation}
with (11.8) $P = \frac{1}{4\pi}(\epsilon-1)E$ (in SGS), the field inside hole:
\begin{equation}
E_{hole} = \frac{\epsilon + 2}{3} E
\end{equation}

Another solution based directly on the superposition pinciple and boundary conditions for the electric field.

Lets look to the sphere's equatorial plane. The field on every point consist of field $E$ - field inside dielectric and a field of dipole $\frac{p}{r^3}$:
\begin{equation}
E_{equator} = E + \frac{p}{\epsilon r^3}
\end{equation}

Boundary conditions for the tangential fields
\begin{equation}
E_{hole} = E_{equator}
\end{equation}

For one of the pole point
\begin{equation}
E_{pole} = E - \frac{2p}{\epsilon r^3}
\end{equation}

Boundary conditions for the normal field on pole
\begin{equation}
E_{hole} = \epsilon E_{pole}
\end{equation}

Thus I have two equations
\begin{align}
E_{hole} = E + \frac{p}{\epsilon r^3} \\
E_{hole}/\epsilon = E - \frac{2p}{\epsilon r^3}
\end{align}

This two equations give me the different answer:
\begin{equation}
E_{hole} = \frac{3\epsilon}{2\epsilon + 1} E
\end{equation}

 

[metatrons]

I believe the first answer is correct, with very solid derivation.

For your second approach, I am not sure about your Equation-3. If I understand correctly, it requires the assumption of a uniform field distribution. However, here the field distribution in the sphere should be non-uniform (distorted polarization in the curvature surface, and more concentrated into poles).

 

[sergiokapone]

metatrons, thanks, but I found the second answer the same as in Jackson "Classical Electrodynamics" (Chapter 4, 4.4 Boundary- Value Problems with Dielectrics, p 159, (4.59)) obtained there in different way.

 

[metatrons]

metatrons, thanks, but I found the second answer the same as in Jackson "Classical Electrodynamics" (Chapter 4, 4.4 Boundary- Value Problems with Dielectrics, p 159, (4.59)) obtained there in different way.

Other than that I did not find anything wrong in both derivation. I would also like to see if there are a sound explanation about it.

 

[nasu]

What do you call E in your eq 3?
In Feynman's treatment, E is the field in the dielectric without a hole. So the field in the equatorial plane of the sphere (in the hole) is E. This is the superposition of the filed without the sphere plus the dipole field of the sphere.

 

[sergiokapone]

In eq 3, $E$ -- is the field inside the dielectric without hole (or the field far from the hole). So, the field in the equatorial plane is supperposition of the $E$ field and field of the hole (which is dipole field). Is something uncorrect here?

 

[nasu]

Then E_equator is the field with the sphere in hole or without?

 

[sergiokapone]

I do not understand question. Hole - is the sphere. Hole - is a spherical hole. Thus E_equator is the field of the spherical hole + field E.