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. $ \frac{ 4 \pi \varepsilon_0 F d^2 }{ e^2}$
• B. $ \sqrt{ \frac{ 4 \pi \varepsilon_0 F e^2 }{ d^2}}$
• C. $ \sqrt{ \frac{ 4 \pi \varepsilon_0 F d^2 }{ e^2}}$
• D. $ \frac{ 4 \pi \varepsilon_0 F d^2 }{ q^2}$

Answer 1

The Correct Option is C

To determine the number of electrons missing from each ion, we start with understanding the force of repulsion between the two ions using Coulomb's Law. The formula for the electrostatic force F between two charges q separated by a distance d is given by:

F = \frac{1}{4 \pi \varepsilon_0} \cdot \frac{q^2}{d^2}

Here, \varepsilon_0 is the permittivity of free space.

We need to find out how many electrons are missing from each ion. Each electron carries a charge of e, so if n electrons are missing, the charge on each ion will be q = ne.

Substitute q = ne into the force equation:

F = \frac{1}{4 \pi \varepsilon_0} \cdot \frac{(ne)^2}{d^2}

Simplifying gives:

F = \frac{n^2 e^2}{4 \pi \varepsilon_0 d^2}

We can rearrange for n^2:

n^2 = \frac{4 \pi \varepsilon_0 F d^2}{e^2}

The number of electrons n is obtained by taking the square root:

n = \sqrt{\frac{4 \pi \varepsilon_0 F d^2}{e^2}}

Thus, the number of electrons missing from each ion is given by this expression, which matches option C: \sqrt{ \frac{4 \pi \varepsilon_0 F d^2}{e^2}}