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.

Options

A. `2q , 4q`
B. `(2q)/(3) , 4q`
C. `(2q)/(3) ,(4q)/(3)`
D. `2q ,(4q)/(3)`

Answer

4

Explanation

`V_(0) = (1)/( 4pi in_(0)) [ (q)/(1) + (q)/(2) + (q)/(4) + … oo]`
` = (q)/(4 pi in_(0)) [ 1 + (1)/(2) + (1)/(2^(2)) + … oo = (1)/((1 - 1//2))]`
`= ((1)/(4 pi in_(0))) 2q`
`E_(0) = (1)/( 4 pi in_(0)) [(q)/(1^(2)) + (q)/(2^(2)) + (q)/(4^(2)) + ...oo]`
`= (q)/(4 pi in_(0)) [ 1 + (1)/(16) + ...oo = (1)/((1 -1//4))]`
`= (q)/(4 pi in_(0)) ((4)/(3)) = (1)/( 4 pi in_(0)) ((4q)/(3))`