Question

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

Answer 1

Let us consider a system of n charges q₁, q₂, ... q_n with position vectors r_1, r₂, r₃, ...r_n relative to origin O.

Let `vecF_i` be the force due to i^th charge q_i on q₀.
Then,

`vecF_i = 1/(4πε_0) (q_1q_0)/(r_i^2) \ hat r_i`

Here, r_i is the distance of the test charge q₀ from q_i.

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`