Two similar spheres each of mass m and carrying equal charges q are hung from a common point by means of two silk threads each of length / Show that the separation x
between them is given by ` x= ((q^(2) hatl)/(2piin_0mg))^(1//3)`
when the angle between the threads is very small.

Explanation

Consider two similar spheres A and B , hung from a common point (O).

Force between these spheres is
` F = (1)/(4piin_0) (q^(2))/(x^(2))" "…(i)`
Let angle made by the string with the verticle is ` theta. `
` therefore T cos theta = mg and T sin theta = F `
` therefore " " tan theta = (F)/(mg) " "...(ii)`
When the angle `theta ` between the threads is very small
` " " 2theta = (x)/(l) rArr theta = (x)/(2l)`
` therefore " " tan theta= theta = (x)/(2l) " "...(iii)`
Now, from Eqs. (i) , (ii) and (iii) , we have
` " " (x)/(2l) = (1)/(mg) xx (1)/(4piin_0) (q^(2))/(x^(2))`
` rArr " " x^(3) = (q^(2)l)/(2 piin_0 mg) `
` rArr " " x = ((q^(2)l)/(2 piin_0mg))^(1//3)`