Question

derive an expression for the electric field at a point on the axis of an electric dipole

Answer 1

     Let the electric dipole be placed such that their center is at the Origin (0,0).  Let the charges be q and -q.  Let "-q" be placed at A(-a, 0) and "q" be placed at B(a, 0).  So x-axis is the axis of the dipole. y-axis is the perpendicular bisector of the dipole.

   Let us take a point P (x, 0) on the axis of the dipole at a distance x from O. The magnitude of electric field at P in the direction of +ve x axis:

1) For x > a:

      E = q/(4πε) * [ 1/(x-a)² - 1/(x+a)² ]
         = q/(4πε) * (4 a x) / [ x² - a² ]²
         = 1/(4πε)  * 2 P x / [x² - a² ]²

        Here P = dipole moment = 2 a q.

2) For -a  < x < a

        E = - q/(4πε) * [ 1/(a - x)² + 1/(a+x)² ] 
           = - q/(4πε) * 2 (a² + x²) / [ a² - x² ]²

       E is in the direction of negative x axis.

3)  For  x < - a:  We can derive it similar to case (1).

       E = - q/(4πε) 2 P x / (x² - a²)²

Answer 2

Explanation:

EXPRESSION IS :

E = 2Kp/X^3.

DERIVATION REFER IN ATTACMENT.

Thank you