Question

Two small spheres each having mass m kg and charge q coulomb are suspended from a point by insulating threads each 1 metre long but of negligible mass. If θ is the angle each string makes with the vertical when equilibrium has been attained, show that 

q² = (4mgl² sin² θ tan θ) 4 πɛ0.

Answer 1

Gravitational as well as electrostatic force act on each sphere.

Consider two small spheres A and B each of mass 

m kg and charge q coulomb. When the two spheres are suspended from point S by two threads each of length I, they repel each other and when equilibrium is attained, each string makes an angle θ with the vertical [See figure.].

Each of the two spheres is acted upon by the following three forces: 

(i) The electrostatic force of repulsion f directed away from each other. 

(ii) The weight mg of the sphere acting vertically downwards. 

(iii) The tension T in the string directed towards point S. Since the two spheres are in equilibrium, the three forces acting on a sphere can be represented by the three sides of the ∆AOS taken in order. For sphere A, we have at equilibrium by Lami’s theorem