Understanding Dipole Behavior in a Dielectric Medium

[XCBRA]

Homework Statement

A dipole p is situated at thecentre of a spherical cavity of radius a in an infiite medium of relative permitivity \epsilon_r. show that the potential in the dielectric medium is the same as would be produced by a dipole p' immersed in a continuous dielectric, where

p'=p\frac{3\epsilon_r}{2\epsilon_r +1}

and that the field strength inside the cavity is equal to that which the dipole would produce in the absence of the dielectric, plus a uniform field E

E=\frac{2(\epsilon_r-1)}{2\epsilon_r + 1}\frac{p}{4\pi\epsilon_0a^3}.

Homework Equations

The Attempt at a Solution

I am not sure at all how to approach this question. I would like to say that I would use spherical harmonics but i am not sure how to apply them in this case.

Would it be possible to say that at large distances

V_2= -\frac{p\cos\theta}{4\pi\epsilon_0\epsilon_r r^2}

then to add then assume that outside the sphere that

V_2= -\frac{p\cos\theta}{4\pi\epsilon_0\epsilon_r r^2} + \frac{A_2\cos\theta}{r^2}

and inisde the sphere that

V_1= B_1 r \cos\theta + \frac{B_2\cos\theta}{r^2}

and then solve the problem using the boundary conditions for tangential E and perpendicular D?

I am really unsure of how to solve this and any help will be greatly appreciated.

 

[mathfeel]

Would it be possible to say that at large distances V_2= -\frac{p\cos\theta}{4\pi\epsilon_0\epsilon_r r^2}then to add then assume that outside the sphere thatV_2= -\frac{p\cos\theta}{4\pi\epsilon_0\epsilon_r r^2} + \frac{A_2\cos\theta}{r^2}

Kind of. Since A_2 is unknown, you might as well just try the potential: V_2 = - \frac{p^{\prime} \cos\theta}{4\pi \epsilon_0 \epsilon_r r^2}, where p^{\prime} is unknown and to be solved.

The whole point of the problem is that at large r, the field looks like something due to some effective dipole moment p^{\prime}, whose value you are to find.

and inisde the sphere thatV_1= B_1 r \cos\theta + \frac{B_2\cos\theta}{r^2}and then solve the problem using the boundary conditions for tangential E and perpendicular D?I am really unsure of how to solve this and any help will be greatly appreciated.

D is okay, but instead of using E, it is easier to use the condition that V is continuous.

So now you have two equations but three unknown: B_1, B_2, and p^{\prime}. But one of them can be found by consider the limit r\to 0.

 

[XCBRA]

Ahh ok that makes a lot more sense, thank you for the help.