Question

Consider a system of n charges q1, q2,…,qn with position vectors 1, 2, 3, …, n relative to some origin ‘O’. Deduce the expression for the net electric field ⃗ at a point P with position vector p, due to this system of charges.

Answer 1

Given: n charges q1, q2,…,qn, position vectors r1, r2, r3, …, rn

To find: Net electric field at a point P.

Solution:

• Consider two point charge of  + q1 and q2  located at point  A and P in space , and position vector for point OA and OP be r1 and r2.

• Now according to triangle law,  from triangle law of vector addition:

               OA + AP = OP

               AP = r12 = OP - OA

               r12 = r2 - r1

• Using Coulomb’s Law,

               F = k x q1 x q2x r(cap) 12/ (r12)²  = k x q1 x q2 x r(cap) 12/ (r12)³  

               F = k x q1 x q2 x (r2 - r1)/mod(r2 - r1)³

• Therefore:

               E = F/Q = kq x (r2 - r1)/mod(r2 - r1)³

• Due to n charges, E would be:

                          n

               E = k x ∑  qi(r - r(i))/ mod(r-r(i))³

                           i=1

Answer:

              The net electric field is    

                                                            n

                                                 E = k x ∑  qi(r - r(i))/ mod(r-r(i))³

                                                              i=1