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Two identical charged spheres hang from a common point on two massless threads of length l and, because of mutual repulsion, settle a small distance x = d apart (d << l). Charge now leaks off both spheres at a constant rate, and the spheres approach each other with speed v. How does v depend on the separation x?

1. v proportional to x^(1/2)
2. v proportional to x
3. v proportional to x^(-1/2)
4. v proportional to x^(-1)

Correct answer: v proportional to x^(-1/2)

Solution

For small separation, equilibrium gives k q² / x² proportional to x (the horizontal restoring force ~ (mg)(x/2l)), so q² is proportional to x³, i.e. q proportional to x^(3/2). Since charge leaks at a constant rate, dq/dt is constant. Differentiating q ~ x^(3/2): dq/dt ~ x^(1/2) (dx/dt). With dq/dt constant, v = dx/dt is proportional to x^(-1/2).

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