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 (e being the charge on an electron).
A) $\dfrac{{4\pi {\varepsilon _o}F{d^2}}}{{{e^2}}}$.
B) $\sqrt {\dfrac{{4\pi {\varepsilon _o}F{e^2}}}{{{d^2}}}} $.
C) $\sqrt {\dfrac{{4\pi {\varepsilon _o}F{d^2}}}{{{e^2}}}} $.
D) $\dfrac{{4\pi {\varepsilon _o}F{d^2}}}{{{q^2}}}$.

Answer 1

Hint: The force between the two charges depends upon the nature of the two charges, if the two charges are positive ions then there is repulsion between the two the charges and if they are negative charge ions then also there will be repulsion only for positive and negative charge ions there is attraction force.

Formula used:
The formula of the Coulomb's law is given by,
$F = k\dfrac{{{q_1} \times {q_2}}}{{{d^2}}}$
Where force is F, the charges are ${q_1}$ and ${q_2}$ also the distance between the charges is d.

Complete step by step solution:
It is given in the problem two positive ions, each carrying a charge q, are separated by a distance d if F is the force of repulsion between the ions then we need to find the number of electrons missing from each ion will be (e being the charge on an electron).
The formula of the Coulomb's law is given by,
$ \Rightarrow F = k\dfrac{{{q_1} \times {q_2}}}{{{d^2}}}$
Where force is F, the charges are ${q_1}$ and ${q_2}$ also the distance between the charges is d.
$ \Rightarrow F = k\dfrac{{{q_1} \times {q_2}}}{{{d^2}}}$
As both the charge, $q = ne$ and the distance is d. Therefore we get,
$ \Rightarrow {q_1} = {q_2} = ne$
$ \Rightarrow F = k\dfrac{{ne \times ne}}{{{d^2}}}$
$ \Rightarrow F = k\dfrac{{{n^2}{e^2}}}{{{d^2}}}$
$ \Rightarrow {n^2}{e^2} = \dfrac{{F \times {d^2}}}{k}$
$ \Rightarrow {n^2} = \dfrac{{F \times {d^2}}}{{k \times {e^2}}}$
$ \Rightarrow n = \sqrt {\dfrac{{F \times {d^2}}}{{k \times {e^2}}}} $.
Replacing the value of constant, $k = \dfrac{1}{{4\pi {\varepsilon _o}}}$ we get,
$ \Rightarrow n = \sqrt {\dfrac{{4\pi {\varepsilon _o} \times {d^2}}}{{{e^2}}}} $.
The number of electrons missing from each item is equal to $n = \sqrt {\dfrac{{4\pi {\varepsilon _o} \times {d^2}}}{{{e^2}}}} $. The nature of force is repulsion as both of the charges are electrons and therefore both charges will repel each other.

Note: The students should understand and remember the formula of coulomb's law as it is very helpful in solving problems like these. The force between two similar natured charges will always be repulsion and the force between two different natured charges will be attraction.