electric field due to a dipole on point lying at line passing through center ofdipole?

Question

i need derivation . this question is placed in www.plus2physics.com but i am unable to open this site

Answer 1

By symmetry, the electric field on the line through the center of the dipole is parallel to the dipole axis (the vertical components cancel out). The field from -q is E_1 = -q/(4πe0r^2) and from +q is E_2 = -q/(4πe_0r^2). The horizontal components of these fields are E_1cosø and E_2cosø where ø is the angle from each charge to the point, and r is the distance from each charge to the point. If the dipole charges are a distance 2d apart (dipole moment of 2*q*d), the y coordinate of the point is √[r^2-d^2], and ø=arctan(y/d). So r=√[y^2+d^2],and cosø= d/√[y^2+d^2. Putting it all together you get

E(y) = -[q/2πe_0] * 1/(y^2+d^2) * d/√[y^2+d^2] = -[q/(2πe_0)]*d*[x^2+d^2]^-1.5

The sign is negative because I assumed the + charge was on the positive x-axis, and the field is directed in the negative direction.

Answer 2

Use vectors.