Question

Two positive ions, each carrying a charge $q$ are separated by a distance $d$. If $F$ is the force of repulsion between the ions, the number of electrons missing from each ion will be (c being the charge of an electron).
A. $\dfrac{{\sqrt F x}}{{3 \times {{10}^3} \times e}}$
B. $\dfrac{{x \times {{10}^9}\sqrt F }}{{9{e^2}}}$
C. $\dfrac{{x \times {{10}^{ - 4}}\sqrt F }}{{9{e^2}}}$
D. $\dfrac{{x \times {{10}^5}\sqrt {F{e^2}} }}{9}$

Answer 1

we will use the force of repulsion using coulomb’s law, in which the distance between the charges is taken as $x$. Also, we know that every ion charge is an integral multiple of an electron, therefore, $q = ne$. Here, $n$ is the number of electrons. We will substitute this value of charge in the formula of force of repulsion to get the desired answer.

Formula used:
The formula of force of repulsion using coulomb’s law is given below
$F = k\dfrac{{{q_1}{q_2}}}{{{x^2}}}$
Here, $F$ is the force of repulsion, $k$ is constant of proportionality, ${q_1}$ and ${q_2}$ are the charges and $x$ is the distance between the charges.
Also, we know that the charge is an integral multiple of an electron which is given below
$q = ne$
Here, $q$ is the charge, $n$ is the number of electrons and $e$ is the electron in the charge.

Complete step by step answer:
We know that, according to Coulomb’s law, the force of repulsion is given by
$F = k\dfrac{{{q_1}{q_2}}}{{{x^2}}}$
Also, we know that the charge is an integral multiple of an electron which is given below
$q = ne$
Therefore, value of both the charges is given below
${q_1} = ne$
${q_2} = ne$
Now, using both the equations to calculate the number of electrons missing from the ion, as show below
$F = k\dfrac{{\left( {ne} \right)\left( {ne} \right)}}{{{x^2}}}$
$\Rightarrow \,F = k\dfrac{{{{\left( {ne} \right)}^2}}}{{{x^2}}}$
$\Rightarrow \,F = 9 \times {10^9}\dfrac{{{{\left( {ne} \right)}^2}}}{{{x^2}}}$
$\Rightarrow \,F{x^2} = 9 \times {10^9}{\left( {ne} \right)^2}$
$\Rightarrow \,{\left( {ne} \right)^2} = \dfrac{{F{x^2}}}{{9 \times {{10}^9}}}$
$\Rightarrow \,{n^2} = \dfrac{{F{x^2}}}{{9 \times {{10}^9} \times {e^2}}}$
$\therefore \,n = \dfrac{{\sqrt F x}}{{3 \times {{10}^3} \times e}}$
Therefore, the number of electrons missing from each ion will be $\dfrac{{\sqrt F x}}{{3 \times {{10}^3} \times e}}$.

Hence, option A is the correct option.

Note: Here, the value of constant of proportionality is $k = \dfrac{1}{{4\pi {\varepsilon _0}}} = 9 \times {10^9}$. Here, we have not taken $4\pi {\varepsilon _0}$ because it will not cancel out in the solution. Always remember that any charge in this world can be represented as an integral multiple of charge of an electron.