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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:

A

v \propto x

B

v \propto x^(-1/2)

C

v \propto x^(-1)

D

v \propto x^(1/2)

Step-by-Step Solution

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