Electric field due to a system of point charges.

Consider a system of N point charges q_{1}, q_{2},.....q_{N}, having position vectors \(\vec r_1,\vec r_2,...\vec r_N,\) with respect to origin O. We wish to determine the electric field at point P whose position vector is \(\vec r\).

According to Coulomb’s law, the force on charge q_{0} due to charge q_{1} is

Where \(\hat r1P\) is a unit vector in the direction from q_{1} to P and r_{1}p is the distance between q_{1} and P.

Hence the electric field at point P due to charge q_{1} is

Similarly, electric field at P due to charge q_{2} is

According to the principle of superposition of electric fields, the electric field at any point due to a group of point charges is equal to the vector sum of the electric fields produced by each charge individually at that point, when all other charges are assumed to be absent.

Hence, the electric field at point P due to the system of N charges is