Two identical spheres of mass m and charge q are suspended from a common point by two threads of the same length l and it is found that threads make an angle `theta` with the vertical when the spheres are in equilibrium . Calculate the value of q.

Explanation

The force acting on the charged particles suspended from the strings is shown in the figure. There will be tension in the wire due to the force of repulsion between the two charges.

At equilibrium , for charge at A, vertical component of tension in the string will be balanced by the weight of the charge .
`T cos theta =mg` …(i)
Where, `cos theta =x/l`
Also, `T sin theta =F_E =1/(4piepsilon_0)(q.q)/(2x)^2`, where `F_E` is the force of repulsion between charges at A and B .
`rArr T sin theta =F_E = 1/(4piepsilon_0)q^2/(2l cos theta)^2`…(ii) (`because x = l cos theta` )
On dividing equation (i) by (ii) , we get
`(sin theta)/(cos theta)=1/(4piepsilon_0)xxq^2/(4l^2 cos^2 theta)xx1/(mg)`
`rArr q^2=16 pi epsilon_0 mg l^2 sin^2 theta xx tan theta `
`q=4l sin theta sqrt(piepsilon_0 mg tan theta)`