Two identical thin rings each of radius R m are kept on the same axis at a distance of R m apart. If the charges on them are 10 C and 5 C respectively. Calculate the work done in moving charge ' \(q\) ' coulomb from the centre of one ring to that of another _________.

(B)

from figure,
\(\begin{aligned} A O^{\prime} & =B O \\ & =\sqrt{R^2+R^2}=\sqrt{2} R\end{aligned}\)
potential at O
\(V _1=\frac{k Q _1}{ R }+\frac{k Q _2}{\sqrt{2} R }\)
potential at \(O ^{\prime}\)
\(V _2=\frac{k Q _1}{\sqrt{2} R }+\frac{k Q _2}{ R }\)
potential difference
\(\begin{array}{l}\Delta V = V _1- V _2 \\ \Delta V=\frac{k\left( Q _1- Q _2\right)}{ R }+\frac{k}{\sqrt{2} R }\left( Q _2- Q _1\right) \\ \Delta V =\frac{k}{ R }\left( Q _1- Q _2\right)\left[1-\frac{1}{\sqrt{2}}\right] \\ \Delta V =\frac{k}{ R }\left( Q _1- Q _2\right)\left[\frac{\sqrt{2}-1}{\sqrt{2}}\right]\end{array}\)
work done \(W =q \Delta V\)
\(\begin{aligned} W & =\frac{k q}{ R }\left( Q _1- Q _2\right)\left[\frac{\sqrt{2}-1}{\sqrt{2}}\right] \\ \therefore W & =\frac{q}{4 \pi \varepsilon_0 R }(10-5)\left[1-\frac{1}{\sqrt{2}}\right] \\ W & =\frac{5 q}{4 \pi \varepsilon_0 R }\left[1-\frac{1}{\sqrt{2}}\right]\end{aligned}\)