2.3 P OTENTIAL DUE TO A P OINT C HARGE

. P OTENTIAL DUE TO A P OINT C HARGE Consider a point charge Q at the origin (Fig. . ). For definiteness, take Q to be positive. We wish to determine the potential at any point P with position vector r from the origin. For that we must calculate the work done in bringing a unit positive test charge from infinity to the point P. For Q > , the work done against the repulsive force on the test charge is positive. Since work done is independent of the path, we choose a convenient path – along the radial direction from infinity to the point P. At some intermediate point P ′ on the path, the electrostatic force on a unit positive charge is ˆ ' Q ′ ( . ) where ˆ ′ r is the unit vector along OP ′ . Work done against this force from r ′ to r ′ + ∆ r ′ is ' Q W ∆ = − ∆′ ( . ) The negative sign appears because for ∆ r ′ < , ∆ W is positive . Total work done (W) by the external force is obtained by integrating Eq. ( . ) from r ′ = ∞ to r ′ = r , ' Q Q Q W dr ∞ ∞ = − ′ ′