Question

Electric charges having same magnitude of electric charge 'q', coulombs are placed at x=1m, 2m, 3m, 4m, 8m, ....... and so on. If any two consecutive charges have opposite sign but the first charge is necessarily positive, what will be the potential at x = O?

• A. $\frac{2hq}{3}$
• B. Zero
• C. $\frac{1}{4\pi {\epsilon }_0}\left(\frac{2q}{3}\right)$
• D. $\frac{1}{4\pi {\epsilon }_O}\left(2q\right)$

Answer 1

The correct option is C $\frac{1}{4\pi {\epsilon }_0}\left(\frac{2q}{3}\right)$

Potential due to a charge at distance $r$ is $\frac{kq}{r}$.

Potential due to the + charge at distance $1m$ is $\frac{kq}{1}$.

Potential due to the -charge at distance $4m$ is $-\frac{kq}{2}$

potential due to the the charge at distance $4m$ is $\frac{kq}{4}$

and so

The total is $kq\left(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+..........\right)$

The thing inside parentheses is a geometeric series. The sum of an infinite geometric senses is a $(1-r)$

where a = first term =1 and r=common ratio=-1/2

So, the infinite sum is $\frac{1}{\left(1-\left(-\frac{1}{2}\right)\right)}=\frac{1}{\left(\frac{3}{2}\right)}=\frac{2}{3}.$

Thus the total potential is $kq\times \frac{2}{3}$