# Electrostatics charged spheres problem

1)Two small equally charged spheres, each of mass m, are suspended from the same point by silk threads of length l.The distance between the spheres x<<l.Find the rate dq/dt with which the charge leaks off each sphere if their approach velocity varies as v=a/(x)^0.5, a is a constant.
The problem can be done by expressing q^2 in equilibrium condition and then differentiating w.r.t. time.But I wonder why the charge leakage will occur?What principle makes it and how?
2)A thin wire ring of radius R has an electric charge q. What will be the increment of the force streatching the wire if a point charge qo is placed at the ring's centre?
It appeared to me that the direct mathematical formulation of E field, inside and outside the ring is rather difficult.so i tried to calculate the potential due to the ring and point charge.But, the integration to find contribution of the ring turned hopelessly difficult.Please help.
Kolahal Bhattacharya.

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1) Due to ions in atmosphere cancelling charges on the spheres. These charged particles can originate from various sources - natural radioactivity in the vicinity/ground/materials in the surroundings.

A larger source of discharge works as follows. Any uncharged particle approaching the sphere will be induced with an opposite charge closest to the sphere. This will attract it to the sphere. Once stuck on it it will gain the same charge as the sphere (the particle was initially neutral). Now it will be repelled by the sphere, flying off into space carrying away some of the charge on the sphere.

Another possible route of discharge happens when the threads absorb moisture out of the atmosphere. This will make them conductive, providing a path for discharge.

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I understand.But could you please clarify, what feature of this problem makes the charged sphere to gradually leak out the charge.In other words, why the charge leakage is necessary in this problem?

Kolahal Bhattacharya said:
I understand.But could you please clarify, what feature of this problem makes the charged sphere to gradually leak out the charge.In other words, why the charge leakage is necessary in this problem?
To be honest with you, this is one of Irodov's artificial sounding problems On-axis field due to a ring is fairly easy to compute. But you don't need it here.

When you place the point charge at the center of the ring, the ring experiences a net outward (repulsive) force. Can you calculate the force on an element of the ring due to the charge and integrate it?

I did it:
dF=k ∫(q0λdl)/r^2
put dl=r dφ
have dF=k ∫(q0λr dφ)/r^2
Then, dF=(kq0λ/r) ∫dφ
integrating o to 2 pi, F=...