Question

Define dipole moment of an electric dipole. show mathematically that the electric field intensity due to a short dipole at a distance d along its axis is twice the intensity at the same distance d along equatorial axis.

Answer 1

The equal and opposite force seprated by some distance is known as dipole .... The dipole moment define as the mathematical product of the separation of the ends of a dipole and the magnitude of the charges...
u

Answer 2

The measure of strength of an electric dipole is known as the dipole moment. It is a vector quantity having magnitude equal to the product of the magnitude of either the charge or the distance between them.

• Potential dipole energy is the energy the dipole absorbs by virtue of its specific position in the electric field.  Suppose an electrical moment dipole P is oriented at an angle with the direction of the uniform external electrical field E as we know, the torque on the dipole is T = pE Sin Ф
• Small amount of work done by rotating dipole by a small angle dФ against torque will be - dW= tdФ = pEsinФdФ

• Total work done while rotating the dipole from orientation Ф1 to Ф2 to is  W=∫pE sin(Ф)dФ = pE ( -cosФ)