Four point charges are placed on the x-axis: +3q at x = -2a , -q at x = -a , +q at x = a , and -3q at x = 2a . The magnitude of the electric field at a point P(0, r) on the y-axis, where r a , is given by [ qa 4 ₀ r^3 ] . The value of is ______.

Step-by-step solution

The system can be viewed as two collinear dipoles. The inner charges -q at x = -a and +q at x = a form a dipole of moment p₁ = q(2a) = 2qa directed along the +x axis. The outer charges -3q at x = 2a and +3q at x = -2a form a dipole of moment p₂ = 3q(4a) = 12qa directed along the -x axis. The net dipole moment is p_ net = 12qa - 2qa = 10qa directed along the -x axis. The point P(0, r) lies on the y-axis, which is the equatorial line of this net dipole. The magnitude of the electric field at an equatorial point at a