Two identicaln thin rings each of radius 10 m carrying charges 10 C and 5 C are coaxially placed at a distance 10 m apart. The work done in moving a charge q from the centre of the first ring to that of the second is

Options

A. `q/(8piepsilon_0)((sqrt2+1)/2)`
B. `q/(8piepsilon_0)((sqrt2+1)/2)`
C. `q/(4piepsilon_0)((sqrt2+1)/2)`
D. `q/(4piepsilon_0)((sqrt2-1)/2)`

Answer

B

Explanation

Potential at the center of the ring 1 is –
`V_1=1/(4piepsilon_0)[q_1/R_1+q_2/(sqrt(R_1^2+d^2))]`
`V_1=1/(4piepsilon_0)[10/10+5/sqrt(10^2+10^2)]`
`=1/(4piepsilon_0)[10/10+5/(10sqrt(2))]`
Similarly
`V_2=1/(4piepsilon_0)[q_2/R_2+q_1/(sqrt(R_2^2+d^2))]`
`=1/(4piepsilon_0)[5/10+10/(10sqrt(2))]`
Work done -
`W=q(V_1-V_2)=q/(4piepsilon_0)[10/10+5/(10sqrt2)-5/10 -10/(10sqrt2)]`
`=q/(4piepsilon_0)[5/10 -5/(10sqrt2)] =q/(4piepsilon_0)[1/2-1/(2sqrt2)]`
`=q/(8pi epsilon_0) [(sqrt2-1)/sqrt2]`