Field inside spherical hole inside dielectric

In summary, there are two different solutions to the problem of finding the electric field inside a dielectric with a small hole. One solution, based on the equations (11.25) and (11.8) from http://www.feynmanlectures.caltech.edu/II_11.html#Ch11-S4, yields an equation for the electric field inside the hole as (epsilon + 2)/3 * E. The other solution, based on the superposition principle and boundary conditions, yields an equation for the electric field inside the hole as 3*epsilon/(2*epsilon + 1) * E. While both derivations seem valid, further explanation is needed to determine which solution is correct.
  • #1
302
17
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}
 
Last edited:
Physics news on Phys.org
  • #2
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).
 
  • #3
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.
 
  • #4
sergiokapone said:
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.
 
  • #5
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.
 
  • #6
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?
 
  • #7
Then E_equator is the field with the sphere in hole or without?
 
  • #8
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.
 

1. What is the concept of a "field inside spherical hole inside dielectric"?

The concept of a field inside spherical hole inside dielectric refers to the electric field that exists within a spherical cavity or hole that is filled with a dielectric material. This electric field is created by the presence of charged particles within the dielectric material.

2. How is the electric field inside a spherical hole inside dielectric calculated?

The electric field inside a spherical hole inside dielectric can be calculated using the formula E = Q/(4πεr^2), where Q is the charge of the particles within the dielectric material, ε is the permittivity of the material, and r is the distance from the center of the spherical hole to the location where the electric field is being measured.

3. What factors affect the strength of the electric field inside a spherical hole inside dielectric?

The strength of the electric field inside a spherical hole inside dielectric is affected by the charge of the particles within the dielectric material, the permittivity of the material, the distance from the center of the spherical hole, and the size and shape of the hole.

4. How does the presence of a dielectric material affect the electric field inside a spherical hole?

The presence of a dielectric material within a spherical hole creates a polarizing effect, which causes the electric field to be stronger inside the hole compared to the surrounding space. This is due to the alignment of the charges within the dielectric material, which enhances the electric field strength.

5. Can the electric field inside a spherical hole inside dielectric be manipulated?

Yes, the electric field inside a spherical hole inside dielectric can be manipulated by changing the charge of the particles within the dielectric material, altering the permittivity of the material, or changing the size and shape of the spherical hole. This makes it a useful concept in various applications, such as in capacitors and electronic devices.

Suggested for: Field inside spherical hole inside dielectric

Back
Top