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A dipole consists of two point charges +q and -q separated by a fixed distance l. The dipole has mass m. It is aligned along the x-axis and enters at speed v0 into a region of electric field of length 2L (L >> l). Inside this region the electric field points along the x-axis and its magnitude varies as E(x) = E0(1 - x²/L²), where x is measured from the centre of the region. The time taken by the dipole to cross this region is (alpha*L)/v0. Find the value of alpha to the nearest integer. [Given: qE0*l/(m*v0²) = 1/6]
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Correct answer: 2
Solution
The dipole experiences a restoring force F = -2qE0*l*x/L², analogous to SHM with angular frequency omega = v0/sqrt(3)/L. With amplitude 2L and entry at x = -L with speed v0, the dipole exits at x = +L after time t = (pi/sqrt(3)) * (L/v0), giving alpha = pi/sqrt(3) ≈ 1.81, nearest integer 2.
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