Two identical thin ring, each of radius R meters, are coaxially placed a distance R metres apart. If `Q_1` coulomb, and `Q_2` coulomb, are repectively the charges uniformly spread on the two rings, the work done in moving a charge q from the centre of one ring to that of the other is

Options

A. zero
B. `(qsqrt(2))/(4pi epsilon_(0)a)(Q_(1) - Q_(2))`
C. `(q(sqrt(2)-1))/(4pi epsilon_(0)asqrt(2))(Q_(1) - Q_(2))`
D. `(q(sqrt(2)-1))/(4pi epsilon_(0)a)(Q_(1) - Q_(2))`

Answer

C

Explanation

The electrostatic potential at the centre of the first ring (i.e., at O) with charge `Q_(1)` is due to charge `Q_(1)` itself as well as due to charge `Q_(2)` on the second ring which is given by
`V_(1) = (1)/(4pi epsilon_(0))(Q_(1))/(a) + (1)/(4pi epsilon_(0))(Q_(2))/(asqrt(2))`
Similarly, the electrostatic potential at the centre of the second ring (i.e., at O.) is given by
`V_(2) = (1)/(4pi epsilon_(0))(Q_(1))/(asqrt(2)) + (1)/(4pi epsilon_(0)) (Q_(2))/(a)`
Required work done, `W = q(V_(1) - V_(2))`
`= q[(1)/(4pi epsilon_(0))(Q_(1))/(a) + (1)/(4pi epsilon_(0))(Q_(2))/(asqrt(2)) - (1)/(4pi epsilon_(0))(Q_(1))/(a sqrt(2))-(1)/(4pi epsilon_(0))(Q_(2))/(a)]`
`= (a)/(4pi epsilon_(0)a)[Q_(1) + (Q_(2))/(sqrt(2)) - (Q_(1))/(sqrt(2)) - Q_(2)]`
`= (q)/(4pi epsilon_(0) asqrt(2))[sqrt(2) (Q_(1) - Q_(2)) - 1(Q_(1) - Q_(2))] = (q(sqrt(2)-1))/(4pi epsilon_(0)asqrt(2))(Q_(1) - Q_(2))`