Electric potential: The electric potential at a point in the electric field due to a charge is the amount of work done to bring a unit positive charge (test charge) from infinity to that point against the electrostatics force of the electric field of source charge.
Units of electrostatic potential:
C.G.S. unit: statvolt – If one erg of work is required to move charge of one statcoulomb from infinity to the given point, against the electrostatic field, then the potential at the given point is 1 statvolt. So, 1 statvolt = 
.
SI unit: volt – If one joule of work is required to move charge of one coulomb from infinity to the given point, against the electrostatic field, then the potential at the given point is 1volt. So, 1 volt = 
.
Relation between volt and statvolt: 1 volt = = 
= 
statvolt.
In C.G.S. unit electromagnetic system, the unit of electrostatic potential is emu of potential or abvolt. So, 1 volt = 
= 108 abvolt.
Potential at a point due to a point charge:
Let us consider a point charge q is placed at point O. To calculate electric potential at a point P at a distance r from O in the electric field of q, we consider a test charge q0 is moved from infinity to point P.
The force acting on q0 when it is at 
moved from infinity is 
= 
———-(i)
To move q0 at constant speed an external force = is applied in opposite direction.
The work done by F to move q0 from P/ to P// (where P/P// = dr/) is dw = – dr/ ——–(ii)
[-ve is due to the work done by external force F to move q0 towards q.]
From equations (i) and (ii) we get, dw = – 
The total work done by external force F to move q0 from infinity to point P is W = 
dw = 
– = – kqq0 
= – kqq0[
– 
] = 
The potential at a point P is V(q0 = 1) = 
= 
= 
.
. Potential energy of a system of three charges:
Let us consider three point charges q1, q2, and q3 are placed at infinity. To calculate potential energy at point A, B and C due to charges q1, q2, and q3 respectively, we have to bring the charges from infinity to those points.
As initially, there is no charge at point A, the potential at point A, VA = 0. So the work done to bring charge q1 from infinity to point A is WA = VA.q1 = 0.
The potential at point B due to q1 is VB = 
.
So the work done to bring charge q2 from infinity to point B is WB = VB.q2 = 
.
The potential at point C due to q1 and q2 is VC = 
+ 
.
So the work done to bring charge q3 from infinity to point C is WC = VC.q3 = 
+ 
The potential energy due to the system of three charges is W = WA + WB + WC
= [ + + ].
Relation between electric potential and electric field:
Let us consider two points P and Q are very close to each other, separated by a distance dr in electric field 
along the direction 
.
The force acting on the test charge q0 placed at Q is = q0 ——(1)
The work done by external force to move q0 from Q to P is dw = – .
——–(2)
[-ve is due to the work done by external force to move q0 against .]
From equations (1) and (2) we get, dw = – q0.
Or, 
= – .
So, dV = – . (the work done dw to move q0 in potential difference dV between points P and Q is dw = q0.dV.)
The electric field is equal to the negative of the potential gradient.
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