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 and is then displaced by a distance x along the line AB . Assuming x d , determine the approximate magnitude of the electric force experienced by particle C .

Step-by-step solution

Let the charges A and B be located at x = - d 2 and x = d 2 respectively. The particle C is displaced to a position x from the origin (midpoint of AB ). The electric force on C due to A is F_A = 1 4 ₀ Q q ( d 2 + x)^2 directed towards B . The electric force on C due to B is F_B = 1 4 ₀ Q q ( d 2 - x)^2 directed towards A . The net force on C is F = F_B - F_A (towards A ): F = Q q 4 ₀ [ 1 ( d 2 - x)^2 - 1 ( d 2 + x)^2 ] F = Q q 4 ₀ [ ( d 2 + x)^2 - ( d 2 - x)^2 ( d^2 4 - x^2)^2 ] F = Q q 4 ₀ [ 2 d x ( d^2 4 - x^2)^2