Question

1. (15 pt) Electric dipole field Oscillating charges produce electromagnetic radiation. The most common antenna is the dipole antenna. Here you will derive the static dipole field at a particular point in space. (a) (8 pt) Find the electric field (magnitude,tq and direction) at point P due to the d2 charges in the figure. Assume free space. d/2 (b) (5 pt) Simplify the expression for the case of rd and write your expression d2 in terms of the magnitude of the electric q dipole moment p-qd (c) (2 pt) Assume the dipole is in a material with a relative permittivity ?, > 1 Is the magnitude of the field at point P larger or smaller than in (b)? Why?

Answer 1

The electric dipole fields due to potential +ve charge (q) at the point 'P' is v_s = q/4 pi c_0 r_1 -(i) +ve charge 9-q) point 'P' is v_2 = -q/4 pi c_0 r_2 = (ii) Hence, 'P_total' is v = v_1 + v_2 = q/4 pi c_0 (1/r_1 -1/r_2) -(iii) Then above values like Fig is From the fig, ab = d/2 cos theta r_1 = r -d/2 cos theta r_2 = r + d/2 cos theta Therefore v = q/4 pi c_0[1/r -d/2 cos theta -1/r + d/a cos theta] -(iv) \r\n Electric field E = -nabla v -(v) from eq(iv) v = q/4 pi c_0 [d cos theta/r^2 -d/2 cos theta] volts -(vi) Electric field E = -nabla v = -[partial differential v/ partial differential r a_x^-+ 1/r partial differential v/ partial differential theta a_theta^-] -(vii) substitute eq (vi) in (vii) we will get the field eq (b) Assuming r > > d: dipolement p = qd