Two charges ' -q ' each are fixed, separated by distance ' 2 d '. A third charge ' q ' of mass ' m ' placed at the mid-point is displaced slightly by ' x ' (x< < d) perpendicular to the line joining the two fixed charges as shown in Fig. The time period of oscillation of ' q ' will be

Step-by-step solution

Let the two fixed charges -q be located at (-d, 0) and (d, 0) . The charge q is at (0, x) . The distance r between the charge q and each of the fixed charges is r = d^2 + x^2 . The force exerted by each fixed charge on q is F = 1 4 ₀ q^2 r^2 . The vertical component of the force from each charge is F_y = F , where = x r . The net restoring force on q is F_ net = -2 F_y = -2 ( 1 4 ₀ q^2 r^2 ) ( x r ) = - q^2 x 2 ₀ r^3 . Since x d , r = d^2 + x^2 d . Thus, F_ net - q^2 x 2 ₀ d^3 . The equation of motion is m d^2x dt^