Two metallic spheres of radii a and b are placed faraway from each other and are connected for a thin conducting wire. Total charge on them is Q. Calculate the potential of each sphere.

(C)
Let potential of sphere having radii a and b are \(V _a\) and \(V _b\) respectively and charge on them be \(Q _a\) and \(Q _b\).
Since two spheres are connected with a wire,
\(V _a= V _b\)
\(\begin{aligned} \therefore \frac{k Q _a}{a} =\frac{k Q _b}{b} \\ \therefore \frac{ Q _a}{ Q _b} =\frac{a}{b} \\ \therefore \frac{ Q _a}{ Q _a+ Q _b} =\frac{a}{a+b} \\ \therefore \frac{ Q _a}{ Q } =\frac{a}{a+b}\end{aligned}\)
\(\left(\because Q _a+ Q _b= Q \right)\)
\(Q _a=\left(\frac{ Q }{a+b}\right) a\)
same as \(\quad Q _b=\left(\frac{ Q }{a+b}\right) b\)
Now, \(V _a=\frac{k Q _a}{a}\)
\(\begin{array}{l} V _a=\frac{k}{a}\left(\frac{ Q }{a+b}\right) a \\ V_a=\frac{k Q }{a+b}= V _b\end{array}\)