Consider a system of n charges q1, q2, ... qn with position vectors vectors r1, r2, r3....rn relative to some origin 'O'.

Question

Consider a system of n charges q₁, q₂, ... q_n with position vectors \(\vec r_1,\vec r_2,\vec r_3....\vec r_n\) relative to some origin 'O'. Deduce the expression for the net electric field \(\vec E\) at a point P with position vector \(\vec r_p,\) due to this system of charges.

Answer 1

 

Electric field due to a system of point charges.

Consider a system of N point charges q₁, q₂,.....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 q₀ due to charge q₁ is

Where \(\hat r1P\) is a unit vector in the direction from q₁ to P and r₁p is the distance between q₁ and P.

Hence the electric field at point P due to charge q₁ is

Similarly, electric field at P due to charge q₂ 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