A question on electricity (potential difference)

[rkrthegreat]

In the situation shown in the figure a particle having charge q is executing simple harmonic motion of amplitude A and time period T in front of a CONDUCTING EARTHED SPHERE of radius R. The reading of the voltmeter when the charge particle is at its mean position(at a distance r from the sphere's centre) is?

The attempt at a solution
I'm confused about the way to approach the problem. There could be so many ways in which the sphere could get a different potential, (like due to the electric field of the charge q, due to changing magnetic field because of the oscillating charge) that I am unable to get a pinpoint solution. Do help me with this one. Look at the attached diagram for clear details.

 

[hikaru1221]

Is there any other information, such as the resistance of the resistor or how fast the particle moves?

 

[rkrthegreat]

Oh, I'm sorry, forgot to put that in the diagram. Take the resistance as R₀. As for how fast the charge moves, the velocity at any position can be calculated, the amplitude of motion and time period being given in the question.

 

[hikaru1221]

I mean, how great is T compared to some other characteristic times, such as r/c (c is speed of light), or the time for charge redistribution? You know, if T is very small compared to them, the problem will be greatly simplified.

 

[rkrthegreat]

Yes, take T as very small

 

[hikaru1221]

Oops sorry. To simplify the problem, T should be very large, not very small. Apology I guess you meant T was large, too?

If T is very large, we may turn the problem to a classical electrostatic one. The field formed by q is electrostatic field. Each position of q corresponds to an electrostatic image on the sphere, which comprises of 2 point charges: one at the center (q₁), the other at somewhere midway between the center and q (q₂). Denote V the potential of the sphere, then q1 is the one responsible for V: V=kq₁/R^2, while q₂ is for canceling the potential on the sphere by q, so q₂ depends on x and q (I don't remember the formulas, but this can found in most books). Besides, the total charge on the sphere is q₁+q₂, and its time derivative is the current through the resistor, and thus: V = R_oI = -R_od(q₁+q₂)/dt. You will arrive at a differential equation, and the rest is solving for V As I remember, the differential equation is quite crazy; but if A<<r, again, we can simplify the equation and it's solvable.

 

[rkrthegreat]

Thanks mate, I guess that's what was to be assumed, to reduce the question to a case of electrostatics. That was what I too needed to confirm, otherwise there could have been many paths to take on this question.
(I wrote take T as small because I thought you were proceeding to solve it using electromagnetics ;-) )