Dipole in a Dielectric: Analysis of r→0

[sachi]

We have a dipole in a vacuum cavity inside an otherwise infinite LIH dielectric.
We assume form Vin = Arcos(theta) + B/(r^2) * cos(theta)
Vout = C/(r^2) * cos(theta)
We are told that "as r tends to 0 the field must approach the dipole field".
I'm not sure if they are talking about the E-field or the potential, as the E-field does not appear to tend to a dipole field as r tends to zero (if you differentiate Vin wrt r to get the E-field, you find a term Acos(theta) which does not tend to zero as r tends to zero. therefore we don't get a purely dipole type field. is it legitimate to let r tend to zero first, then perform the differentiation to get the E-field, or are they just talking about the potential anyway in the first place?)

thanks for your help

 

[Physics Monkey]

The order of differentiation doesn't matter. Remember that the dipole part blows up at r = 0 so the constant term gets completely swamped, that's what they mean when they say the field approaches the pure dipole field. The purpose of this condition is simply to tell you what the coeffecient B is.