Question

Two identical charged spheres suspended from a common point by two massless strings of length l are initially at a distance 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 distance x between the spheres as:

• A. \(v \propto x^{1/2}\)
• B. \(v \propto x\)
• C. \(v \propto x^{-1/2}\)
• D. v ∝ x⁻1

Hint:

Use small-angle approximation and equilibrium conditions for charged pendulums.

Answer 1

The Correct Option is C

Step 1: For small angles, equilibrium gives:
\( \dfrac{k q^2}{x^2} \propto x \)
\( \Rightarrow q^2 \propto x^3 \) 

Step 2: Since charge leaks at a constant rate:
\( q \propto t \)
\( \Rightarrow x^{3/2} \propto t \) 

Step 3: Velocity:
\( v = \dfrac{dx}{dt} \propto x^{-1/2} \)