Two identical charged spheres suspended from a common point by two massless strings of lengths l are initially

Question

Two identical charged spheres suspended from a common point by two massless strings of lengths l are initially at a distance d (d << l) apart because of their mutual repulsion. The charges begin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity v. Then v varies as a function of the distance x between the spheres, as:

Accepted Answer

1. Equilibrium Condition: For a small angle θ (since d << l), tan θ ≈ sin θ = x/(2l). The electrostatic repulsion F_e balances the horizontal component of tension, and gravity mg balances the vertical. Thus, F_e = mg tan θ.
2. Force Equation: (k * q^2) / x^2 = mg * (x / 2l).
3. Relation between q and x: Rearranging gives q^2 ∝ x^3, or q ∝ x^(3/2).
4. Differentiation: Since charge leaks at a constant rate (dq/dt = constant), differentiate q ∝ x^(3/2) with respect to time: dq/dt ∝ (3/2) * x^(1/2) * (dx/dt).
5. Velocity Relation: Since dx/dt = v and dq/dt is constant, we get constant ∝ x^(1/2) * v. Therefore, v ∝ x^(-1/2).

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