Derive an expression for the electric …

Let P be an axial point at distance r from the center of the dipole. Electric potential at point P is given as

V=V1+V 2

V1 and V 2

are the potentials at point P due to charges +q and –q respectively

V=1/ 4πε0( q/r−2a+(-q/ r+2a ))

V= 1/4πε0 ( 4a/(r²-4a²))

V= 1/4πε0(2P/(r²-4a²))

Hence, the electric potential is 1/4πε0 (2P/r²-4a²))

Consider, two charges q and -q seperated by distance 2a such that, it's total charge is 0.

 
Potential due to charge q = 
Potential due to charge -q =  
Potential due to dipole, is the sum of potential due to charges q and -q.
Therefore,
                    
 where, r₁ and r₂ are distances of charges q and -q from point P.

Now, 
 

Considering r much greater than a, r>>a.

Therefore,


Putting these values in the above equation of potential we have,

 

where, 

For potential at any point on axis,  
V= 

Potential is positive when  and potential is negative when  .

Electrical potential falls off at large distance, as  and not as characteristic of the potential due to a single charge.