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Consider a system of n charges q1, q2, ... qn 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.

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Electric field due to a system of point charges.

Consider a system of N point charges q1, q2,.....qN, 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

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