Question

An infinite number of electric charges each equal to $2\,nano\, coulombs$ in magnitude are placed along x-axis at $x = 1\,cm$, $x = 3\,cm$, $x = 9\,cm$, $x = 27\,cm$ …. And so on. In this setup if the consecutive charges have opposite sign, then the electric potential at $x = 0$ is
A. $1250\,V$
B. $1350\,V$
C. $2700\,V$
D. $2500\,V$

Answer 1

In this question, we need to find the electric potential at $x = 0$. We will use the formula of the electric potential at any point around a point charge $Q$. And evaluate the $r$, then we will apply the values and evaluate to determine the required solution.

Complete step by step answer:
The electric potential energy is the amount of work needed to move a unit of electric charge from a reference point to a specific point in an electric field without producing acceleration.Now, we know that the formula of the electric potential at any point around a point charge $Q$ is given by,
$V = k\left[ {\dfrac{Q}{r}} \right]$
Where $V$ is the electric potential energy,$Q$ is a point charge, $r$ is the distance between any point around the charge to the point charge and $k$ is Coulomb constant; $k = 9.0 \times {10^9}\,N$.
$V= kQ\left( {\dfrac{1}{{{r_1}}} + \dfrac{1}{{{r_2}}} + \dfrac{1}{{{r_3}}} + \dfrac{1}{{{r_4}}}...} \right)$
$\Rightarrow V = kQ\left( {1 + \dfrac{1}{3} + \dfrac{1}{9} + \dfrac{1}{{27}}...} \right)$
$\Rightarrow V = kQ\left( {1 + \dfrac{1}{3} + \dfrac{1}{{{3^2}}} + \dfrac{1}{{{3^3}}}...} \right)$
$\Rightarrow V = kQ\left( {\dfrac{1}{{1 - \dfrac{1}{3}}}} \right)$
$\Rightarrow V = kQ \times \dfrac{3}{2}$
Therefore at $x = 0$, the electric potential is,
${V_1} = 9 \times {10^9} \times 2 \times {10^{ - 9}} \times \dfrac{3}{2}$
$\Rightarrow{V_1} = 18 \times \dfrac{3}{2}$
$\Rightarrow {V_1} = 2700\,V$
It is also given that the consecutive charges have opposite signs.
Therefore, the electric potential $= \dfrac{{{V_1}}}{2}$
By substituting ${V_1} = 2700\,V$,
$\Rightarrow\dfrac{{2700}}{2}$
$\Rightarrow 1350\,V$

Hence, option B is the correct answer.

Note: In this question, it is important to note here that, we may think ${V_1} = 2700\,V$ is the required solution and stop upto it but in the question it is also given the consecutive charges have opposite signs so we need divide by $2$ in order to get the required electric potential. So, read the given carefully while solving these types of questions.