21. Obtain an expression for the electric potential at a point which is at a distance r from the point charge Q.

Consider an electric field due to point charge Q placed at point O. Suppose V_A is the electric potential at point A, which is at a distance of r_a from charge Q.

It W_{∞ A} is the work done in moving a small test charge Q₀ from ∞ to A then potential is, 

                         ... (1)
Consider that at any time, the test charge is at point P. The electric field E due to the charge Q exerts force,

 on the test charge placed at that point P. 
If the test charge Q₀ is moved a small distance PB = dl, then small work done is given by, 

dW = F. dl

       = - (Q_oE). dl

       = -Q_o E dl cos 180^o

       =  Q_oEdl

 dW = - Q_oEdr

Here we have taken dl as - dr, because the distance r decreases in the direction of dl. 

Since, we know that 

 

Using equation (2) in eqn. (1), we get