Question

Five equal point charges with $q=10 \, nC$ are located at $x=2$ , $4$ , $5$ , $10$ and $20 \, m$ . Taking the value of permittivity of free space $\epsilon _{0}=\frac{1}{36 \pi }\times 10^{- 9} \, C^{2} \, N^{- 1} \, m^{- 2}$ and potential at infinity to be zero, calculate the potential (in $\text{V}$ ) at $x=0$ ?

Answer 1

Given $q=10 \, nC=​10\times 10^{- 9}C$
The potential at the point $x=0$ , will be the sum of potential produced by the charges placed at $x=2, \, 4, \, 5, \, 10$ and $20m$
Potential,
$V=\frac{q}{4 \pi \epsilon _{0} \times r}$
Then, $V_{1}=\frac{q}{4 \pi \epsilon _{0} \, \times \, 2}$
$V_{2}=\frac{q}{4 \pi \epsilon _{0} \times 4}$
$V_{3}=\frac{q}{4 \pi \epsilon _{0} \times 5}$
$V_{4}=\frac{q}{4 \pi \epsilon _{0} \times 10}$
$V_{5}=\frac{q}{4 \pi \times 20}$
The resultant potential,
$V=V_{1}+V_{2}+V_{3}+V_{4}+V_{5}$
$=\frac{q}{4 \pi \epsilon _{0}}\left[\frac{1}{2} + \frac{1}{4} + \frac{1}{5} + \frac{1}{10} + \frac{1}{20}\right]$
$=\frac{q​}{4 \pi \epsilon _{0}}\left[\frac{10 + 5 + 4 + 2 + 1}{20}\right]$
$=\frac{1}{4 \pi \epsilon _{0}}\times \frac{10 \times 10^{- 9} \times 22}{20}$
$=\frac{1 \times 36 \pi }{4 \pi \times 10^{- 9}}\times \frac{10 \times 10^{- 9} \times 22}{20}$
$=99 \, V$