Question

An infinite number of electric charges equal to $1\text{C}$ in magnitude are placed along $x$-axis at $x=2\text{cm},4\text{cm},8\text{cm}........$so on. If they are of same sign, then electric potential at $x=0$ is

• A. Zero
• B. $1.25\times {10}^9\text{V}$
• C. $3\times {10}^9\text{V}$
• D. $9\times {10}^{11}\text{V}$

Answer 1

The correct option is D $9\times {10}^{11}\text{V}$

Give, all are identical charges having charge $q=1\text{C}$

Potential at the origin is given by,



$V=V_1+V_2+V_3+.......$

$=\frac{q}{4\pi {\epsilon }_0}[\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}.......]$

$=9\times {10}^9\times 1[\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+.......]\times {10}^2$

$=\frac{9\times {10}^9}{2}\times [1+\frac{1}{2}+\frac{1}{2^2}+.......]\times {10}^2$

$=4.5\times {10}^{11}\left[\frac{1}{1-\frac{1}{2}}\right]$

$=9\times {10}^{11}\text{V}$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, (d) is the correct answer.

Note :

$\because [1+\frac{1}{2}+\frac{1}{2^2}+.......]$ is like G.P with infinite terms, its sum is equal to $S=\frac{a}{1-r}$

Where $a=$ First term in the series and $r=$ common factor