The force acting on a charged particle due to electric and magnetic fields is expressed as F = q E + q v B . Suppose E is directed along the X-axis and B is along the Y-axis. In what direction and with what minimum speed v must a positively charged particle be projected so that the net force on it is zero?

Step-by-step solution

Given E = E i and B = B j . The net force on the particle is zero, so F = q E + q( v B ) = 0 . E = -( v B ) Let v = v_x i + v_y j + v_z k . v B = (v_x i + v_y j + v_z k ) (B j ) = v_x B k - v_z B i Substituting this into the force equation: E i - v_z B i + v_x B k = 0 Comparing the components, we get: E - v_z B = 0 v_z = E B v_x B = 0 v_x = 0 The component v_y can be any real number. The speed of the particle is v = v_x^2 + v_y^2 + v_z^2 = v_y^2 + ( E B )^2 . For the minimum speed, v_y = 0 . Thus, v = E B k . The p