Two identical thin rings, each of radius R m, are kept on the same axis at a distance of R m apart. The charges on them are 10 C and 5 C respectively. Calculate the work done in moving a charge q C from the centre of one ring to that of the other.

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(2) (5q)/(4πε₀R)[1-1/(√2)] J. Given: rings of radius R, a distance R apart on a common axis, with Q₁=10 C and Q₂=5 C; write K=1/(4πε₀). Idea: every point of a ring is R from its own centre and √R²+R²=√2R from the other ring's centre, so each potential is a simple sum. Centre of ring 1: V₁=(KQ₁)/R+(KQ₂)/(√2R)=K/R(10+5/(√2)). Centre of ring 2: V₂=(KQ₂)/R+(KQ₁)/(√2R)=K/R(5+(10)/(√2)). V₁-V₂=K/R(5-5/(√2))=(5K)/R(1-1/(√2)). Carrying q from the centre of ring 2 to that of ring 1 takes W=q(V₁-V₂)=(5q)/(4πε₀R)[1-1/(√2)] J; the reverse trip gives the same amount with a minus sign.

Two identical thin rings, each of radius R m, are kept on the same axis at a distance of R m apart. The charges on them are 10 C and 5 C respectively. Calculate the work done in moving a charge q C from the centre of one ring to that of the other.

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