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Question
Consider a system of n charges q1, q2, ... qn with position vectors `vecr_1,vecr_2,vecr_3,...... vecr_n`relative to some origin 'O'. Deduce the expression for the net electric field`vec E` at a point P with position vector `vecr_p,`due to this system of charges.
Solution
Let us consider a system of n charges q1, q2, ... qn with position vectors r1, r2, r3, ...rn relative to origin O.
Let `vecF_i` be the force due to ith charge qi on q0.
Then,
`vecF_i = 1/(4πε_0) (q_1q_0)/(r_i^2) \ hat r_i`
Here, ri is the distance of the test charge q0 from qi.
The electric field at the observation point P is given by `vecE_i = lim_(q0->0) vecF_i/(q_0) = lim _(q0->0) 1/q_0 (1/(4πε_0) (q_1q_0)/r_1^2 hatr_i)`
`vecE_i = 1/(4πε_0) q_i/(r_1^2)hat r_i ........ (1) `
If `vecE` is the electric field at point P due to the system of charges, then by the principle of superposition of electric fields,
`vecE = vecE_1 +vecE_2 +vecE_3 +....vecE_n = sum_(i=1)^n vecE_i`
Using (1), we get
`vecE = sum_(i=1)^n 1/(4πε_0)q_i/(r_i^2)hat r_i `
`vecE =1/(4πε_0) sum_(i=1)^n q_i/(r_i^2)hat r_i`
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