"A dipole in a dielectric: Intriguing results and shape dependence of the distant electric field-.
"A dipole in a dielectric-, a familiar statement to say the least! It was not the mention of dipole that persuaded me to include this article in my portfolio, but the conflicting results attained by two seemingly "correct- methods of determining the dipole moment. I can't say that I enjoy conflict, in fact I don't. I do however realize that many of the giant leaps in physics materialized do to inconsistantcies in what was once accepted as true. In these instances many basic truths needed to be modified to explain the disagreement. Could this be the beginning of something big?.
Consider a dipole consisting of only +q and -q a distance "d- apart. If we let dà 0 then we have a point dipole p0. If we are located in a linear dielectric gauss' law becomes = q. The medium is not a vacuum and the charges in the dielectric can be taken into consideration in the form . Likewise the dipole takes shape as . This is the standard text book solution. Using this information the potential is found to be .
If we approach the problem from a different perspective, using a spherical version of Laplace's equation for azimuthal symmetry and inserting the dipole into an empty spherical cavity of radius R (within the dielectric) and letting RÃ 0, we get a potential in the vacuum region of . The potential in the dielectric region is . By exploiting the continuity of the scalar potential and finding the -field we get and . Applying boundary conditions gives = so that . .
Solving gives:.
, , .
And a different dipole moment than found through conventional methods.
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So what is different? The classic method (text book solution) for solving for the dipole moment was to have two charges in a dielectric and a vacuum cavity between them that shrinks to 0. Method two inserts both charges into the vacuum cavity and them tends the radius of the cavity to 0.