Let each charged sphere have a charge q, have a mass m and make an angle θ with the vertical.
In the initial position of equilibrium electrical force of repulsion F and weight of each sphere is connected through expression
Fmg=tanθ
Using Coulomb's Law and other values we write this as
keq2d2mg=d2(l2−(d2)2)12
⇒q=⎛⎜
⎜
⎜
⎜
⎜⎝mg2ked3(l2−(d2)2)12⎞⎟
⎟
⎟
⎟
⎟⎠12
As the charge starts leaking let x be instantaneous distance between the spheres. The above expression reduces to
q=√mg2kex32(l2−(x2)2)14
Differentiating both sides with respect to time t we get
dqdt=√mg2keddt⎡⎢
⎢
⎢⎣x32(l2−(x2)2)14⎤⎥
⎥
⎥⎦
⇒
dqdt=√mg2ke⎡⎢
⎢
⎢⎣32x12(l2−(x2)2)14+(x32)(−14)(−x2)(l2−(x2)2)−54⎤⎥
⎥
⎥⎦dxdt
dqdt is rate of leakage of charge and velocity dxdt=v. Above expression becomes
1vdqdt=√mg2ke⎡⎢
⎢
⎢⎣3x122(l2−(x2)2)14+x528(l2−(x2)2)−54⎤⎥
⎥
⎥⎦
is the required expression.
