Two particles A and B , each carrying a charge Q , are held fixed with a separation d between them. A particle C having mass m and charge q is kept at the midpoint of the line AB . The particle C is displaced by a small distance x d along the line AB and released. Assuming the charges Q and q have the same sign so that particle C executes simple harmonic motion, determine the time period of the oscillations.

Step-by-step solution

Let the origin be at the midpoint of the line joining A and B . The coordinates of A and B are -d/2 and d/2 . When particle C is displaced by a small distance x towards B , its distance from A becomes d/2 + x and from B becomes d/2 - x . The net force on C is: F = 1 4 ₀ Qq (d/2 + x)^2 - 1 4 ₀ Qq (d/2 - x)^2 F = Qq 4 ₀ [ (d/2 - x)^2 - (d/2 + x)^2 (d^2/4 - x^2)^2 ] F = Qq 4 ₀ [ -2dx (d^2/4 - x^2)^2 ] Since x d , the term x^2 can be neglected in the denominator: F Qq 4 ₀ [ -2dx (d^2/4)^2 ] = - 8Qq ₀ d^3 x Comparing th