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 ($\varepsilon_0$ = permittivity of free space, $e$ = charge on an electron)

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

Hint:

Perform a quick dimensional check! Since $n$ is a pure dimensionless number, everything inside the square root must cancel out structurally. Knowing that $F \propto \frac{q^2}{d^2} \rightarrow q^2 \propto Fd^2$, the term inside the radical simplifies to $\frac{q^2}{e^2} = n^2$, confirming that the formula must be under a square root block.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Two identical positive ions experience an electrostatic repulsive force due to their charges. The positive charge arises because electrons are missing. We need to find the number of missing electrons ($n$) in terms of force, distance, and fundamental constants.

Step 2: Key Formula or Approach:
According to Coulomb's Law, the electrostatic force $F$ between two point charges separated by a distance $d$ in a vacuum is:
$$F = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{d^2}$$ Since both ions carry the same charge $q_1 = q_2 = q$:
$$F = \frac{1}{4\pi\varepsilon_0}\frac{q^2}{d^2}$$ By the quantization of charge, the net positive charge on an ion missing $n$ electrons is:
$$q = ne$$

Step 3: Detailed Explanation:
Substitute $q = ne$ into Coulomb's law equation:
$$F = \frac{1}{4\pi\varepsilon_0}\frac{(ne)^2}{d^2}$$ $$F = \frac{1}{4\pi\varepsilon_0}\frac{n^2 e^2}{d^2}$$ To isolate $n^2$, multiply both sides by $4\pi\varepsilon_0 d^2$ and divide by $e^2$:
$$n^2 = \frac{4\pi\varepsilon_0 F d^2}{e^2}$$ Take the square root of both sides to solve for the number of electrons $n$:
$$n = \sqrt{\frac{4\pi\varepsilon_0 F d^2}{e^2}}$$

Step 4: Final Answer:
The expression for the number of missing electrons is $\sqrt{\frac{4\pi\varepsilon_0 Fd^2}{e^2}}$, which matches option (D).