An infinite number of charges, each of value q, are placed on the x-axis at the points x = 1 , x = 2, x = 4 , x = 8, ………. . Find the potential and intensity due to these charges at x = 0. If the charges are alternately positive and negative what will be the potential and intensity at the same point?

Explanation

If V be the potential at x = 0, then
`V = q/1 + q/2 + q/4 + q/8 + ……..oo`
`= q (1+1/2 + 1/4 + 1/8 + ………..oo) = q = (1/(1-1/2)) = 2q`
If E be the intensity at x = 0, then
`E = q/(1^2) + q/(2^2) + q/(4^2) + q/(8^2) + ………oo`
`q = (1+1/4 + 1/16 + 1/64 + .........oo)`
`=q (1/(1-1/4))=(4q)/3`, in the direction of negative x-axis.
In the second case, when the consecutive charges are of opposite signs, let us assume that the first charge is positive. Then potential,
`V = q/1 - q/2 + q/4 - q/8 + .......oo`
` = q(1-1/2+1/4+1/8+......oo) = q (1/(1+1/2)) = (2q)/3`
Intensity, `E = q/(1^2) - q/(2^2) + q/(4^2) -q/(8^2) + .....oo`
`=q (1-1/4 + 1/16 -1/64 + ........oo)`
`q(1/(1+1/4)) = (4q)/5`, in the direction of negative x-axis.