Question

An infinite number of electric charges each equal to 2 nano coulombs in magnitude are placed along x-axis at x=1 cm, x= 3 , x= 9 cm , x= 27 cm ...... and so on. In these setup if the consecutive charges have oppsoite sign, then the electric potential at x = 0 will be?

Answer 1

The distance of charges from origin follow a geometric series $(d_k=3^{k}cm)$

common ratio = $3^k$

$+a$ $-a$ $+a$ $-a$ .... $+a$ ...... $\infty \left|q\right|=2nc$

$\dot{0}$ $\dot{1}cm$ $\dot{3}cm$ $\dot{27}cm$ $\dot{3^k}cm$

potential due to charges,

$v_1=kq\sum _0^{\infty }(\frac{1}{3}{)}^{2k}$

Here series follows, 1,9,81... common ratio = $(\frac{1}{3}{)}^2=\frac{1}{9}$

$=9\times {10}^9\times 2\times {10}^{-9}\times \frac{1}{1-\frac{1}{9}}$

[sum of infinite $GP=\frac{a_0}{1-r}]$

$\Rightarrow v_1\simeq 1600v$

(remember converting cm to m for d)

and due to $-q$ charges,

$v_2=-kq\sum _1^{\infty }(\frac{1}{3}{)}^{2k-1}$ series follows 3,27,...

$=-3\times 9\times {10}^9\times 2\times {10}^{-9}\times \frac{\frac{1}{3}}{1-\frac{1}{9}}=-1600v$

$\therefore$ Net potential at center, $v_{net}=v_1+v_2=ov$