A conducting sphere 10 cm in diameter is first charged and left isolated at a po

Question

A conducting sphere 10 cm in diameter is first charged and left isolated at a potential of 100 Volts. The sphere is connected afterwards to the ground through a capacitor of capacitance C=10^-12 F. (The capacitor is far from the sphere, therefore the sphere can still be considered to be isolated (i.e., its potential is kQ/a)). Calculate the charge (in Coulombs) on the sphere and on the capacitor after being connected.
Picture: Sphere with line going through capacitor and to the ground. Radius of the sphere is r and the distance between the sphere and capacitor is d. When the sphere and capacitor are connected and are far apart, d >> r.

Explanation / Answer

First calculate the initial charge on the sphere "qo" from the given voltage "Vo" as;
qo = aVo/k

When you connect the sphere to the capacitor plate, charge will flow onto the plate from the sphere until they are both at the same potential, since they become a single conductor.
So you can equate the capacitor voltage to the new sphere voltage;
q1/C = kq2/a

Also, no charge is lost in the process so the capacitor charge "q2" plus the sphere's new charge must add up to the original charge, found above;
q1 + q2 = qo

The last two eqs are two eqs in q1 & q2 which you solve simultaneously by standard algebra to find first one then the other

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