Interactions between a dipole and a point charge

[bumdass]

After solving a homework problem, I realized I don't know what to do when there's a dipole and a point charge but the distance from the charges in the dipole is greater than the distance from the center of the dipole to the charge. As my homework problem stated, with a little context added, "Assume that x [the distance from a point charge to the center of a dipole] is much larger than the separation d between the charges in the dipole, so that the approximate expression for the electric field along the dipole axis can be used."

But what if x was less than d? Now, how would one go about finding the electric field? If you're given the magnitude of the torque (τ), the dipole moment, and the angle (θ) that the electric field lines make with the imaginary line between the point charges of the dipole, then I believe E=τ/(psin(θ)), but is that correct?

Furthermore, how would I find p to plug into the above equation? My prof explained that p=qd, but would d be the distance between the two point charges of the dipole? Also, what if you didn't even have τ or θ? My professor showed us something like this, but I believe he said it only works when x or z is much less than d (i.e. the dipole's point charges are much closer together than the dipole is from the other point charge)

Any links or explanations would be appreciated.

 

[vanhees71]

If you are too close to the two charges, you have to treat them as such, because the dipole approximation becomes inaccurate. If it's a static problem you can just use their electric field
$$\vec{E}(\vec{x})=\frac{q_1 (\vec{x}-\vec{x}_1)}{4 \pi |\vec{x}-\vec{x}_1|^3} + \frac{q_2 (\vec{x}-\vec{x}_2)}{4 \pi |\vec{x}-\vec{x}_2|^3}.$$