An infinite number of charges each equal to q, are placed along the X-axis at `x = 1, x = 2, x = 4, x = 8`,…….. and so on.
(i) find the electric field at a point `x = 0` due to this set up of charges.
(ii) What will be the electric field if the above setup, the consecutive charges have opposite signs.

Explanation

(a) At `O`, direction of electric field by point charges will be towards left.
Electric field at `O`:
Due to `Q at A , E_(1) = (1)/( 4 pi in_(0)) . (Q)/(r^(2))` , towards left
Due to `Q at B , E_(2) = (1)/( 4 pi in_(0)) . (Q)/((2 r)^(2))` , towards left
Due to `Q at C , E_(3) = (1)/( 4 pi in_(0)).(Q)/((4 r)^(2))` , towards left and so on .
Resultant field at `O`:
`E_(0) = E_(1) + E_(2) + E_(3) + .... oo`
`= (1)/(4 pi in_(0)) [(Q)/(r^(2)) + (Q)/((2r)^(2)) + (Q)/((4 r)^(2)) + ...oo]`
`= (1)/(4 pi in_(0)) (Q)/(r^(2)) [ 1 + (1)/(4) + (1)/(16) + ...oo]`
`= (1)/( 4pi in_(0)) .(Q)/(r^(2)) ((1)/(1 - (1)/(4)))`
` = (4)/(3) . (1)/( 4 pi in_(0)) (Q)/(r^(2))` , towards left
(b) The direction of electric field by each `Q` will be towards left and by each `-Q` will be towards right. Electric field by `Q` at `A` will be greater than `-Q` at `B` and hence net field at `O` will be towards left.
Resultant field at `O`:
`E_(0) = (1)/( 4 pi in_(0)) [ (Q)/(r^(2)) - (Q)/((2r)^(2)) + (Q)/((4r)^(2)) - .... oo]`
`E_(0) = (1)/( 4 pi in_(0)) .(Q)/(r^(2)) [ 1 - (1)/(4) + (1)/(16) - ... oo]`
`= (1)/(4 pi in_(0)) .(Q)/(r^(2)) [ (1)/(1 - (-(1)/(4)))]`
`= (4)/(5) .(1)/(4 pi in_(0)) .(Q)/(r^(2))` , towards left