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A spring of spring constant k and negligible natural length is fixed at the origin; a point particle of mass m and positive charge q is attached to its free end on a smooth horizontal surface. A point dipole p, pointing toward q, is fixed at the origin. The spring stretches to length L at the new equilibrium. The mass is displaced slightly by dL (<< L) and released, oscillating with frequency (1/delta)*sqrt(k/m). Find delta.

1. delta = 3.14
2. delta = 2.00
3. delta = 1.00
4. delta = 0.50

Correct answer: delta = 3.14

Solution

The attractive dipole force on q is F_d = (1/(4*pi*eps0))*2pq/r³ (toward origin). At equilibrium k*L = 2pq/(4*pi*eps0*L³), so 2pq/(4*pi*eps0) = k*L⁴. The net restoring stiffness for small displacement is k_eff = k + d/dr[2pq/(4*pi*eps0*r³)] evaluated at L = k + 3*(2pq/(4*pi*eps0))/L⁴ = k + 3k = 4k. Frequency = (1/2pi)*sqrt(4k/m) = (1/pi)*sqrt(k/m), so delta = pi = 3.14.

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