Question

An infinite number of charges, each of charge $1\mu C$ are placed on the x-axis with coordinates \[x=1,2,3\ldots \ldots ..\infty \]. If a charge of \[1C\] is kept at the origin, then when what is the net force acting on \[1C\] charge:
\[\begin{align}
  & A.9000N \\
 & B.12000N \\
 & C.24000N \\
 & D.36000N \\
\end{align}\]

Answer 1

Hint: Electrostatic force or coulomb’s inverse square law, which measures the forces between two charges, which at a distance $r$ from each other. Then the force is proportional to product of the charge and inversely proportional to the square of the distance between them. To remove the proportionality, a quantity $k$ is introduced.

Formula used:
$F=\dfrac{1}{4\pi\epsilon_{0}}\Sigma \dfrac{q_{i}q_{j}}{r_{ij}^{2}}$

Complete answer:
The electrostatic force or the coulomb’s inverse square law was found by the French physicist, named Charles Augustin de Coulomb in the year 1785.
Let $q_{1}$ and $q_{2}$ be two point changes at a distant $r_{12}$ between them, then we know that force due to a point charge is then given by: $F_{12}=\dfrac{1}{4\pi\epsilon_{0}}\dfrac{q_{1}q_{2}}{r_{12}^{2}}$.

If the force of interaction between the charges is attractive, then the value of $F$ is +ve and if the force of interaction between the charges is repulsive, then the value of $F$ is -ve .

Similarly, if $n$ number of charges is present, then the force due to them is given by:

$F=\dfrac{1}{4\pi\epsilon_{0}}\Sigma \dfrac{q_{i}q_{j}}{r_{ij}^{2}}$, where $q_{i},q_{j}$ are the charges , $r_{ij}$ is the distance between them respectively and $\epsilon_{0}$ is the permittivity of vacuum.
We can write $\dfrac{1}{4\pi\epsilon_{0}}=k=9\times10^{9}$ and is also known as coulomb’s constant.
Here, it is given that $q_{1}=1\mu C$ and the $r_{ij}=1,2,3..\infty$, then the $F$ on $q_{1}$, is given by:
$F=k\times10^{-6} \times\left[\dfrac{1}{1^{2}}+\dfrac{1}{2^{2}}+\dfrac{1}{3^{2}}+\dfrac{1}{4^{2}}+..\dfrac{1}{\infty^{2}}\right]$
Or, $F=9\times 10^{9}\times 10^{-6} \times\left[\dfrac{1}{1}+\dfrac{1}{4}+\dfrac{1}{9}+\dfrac{1}{16}+..\right]$
Using the formula sum of infinite series, i.e. $\left(\dfrac{a}{1-r}\right)$, here $a=1$ and $r=\dfrac{1}{4}$
We get, $F=9\times 10^{3}\left[\dfrac{1}{1-\dfrac{1}{4}}\right]$
Or, $F=9\times 10^{3}\left[\dfrac{4}{3}\right]=12000N$
Hence the answer is \[12000N\]

So, the correct answer is “Option B”.

Note:
The coulomb’s inverse square law is valid only for charges in rest. Also, the law can be extended to any number of charges. Note that $F$ is a vector quantity, i.e. it has both magnitude and direction. The resultant force is due to the superposition of forces due to charges, which is done by vector addition.