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)

Options

A. `(4 pi in_(0) Fd^(2))/(q^(2))`
B. `(4pi in_(0) Fd^(2))/(2)`
C. `sqrt((4 pi in_(0) Fe^(2))/(d^(2)))`
D. `sqrt((4pi in_(0) Fd^(2))/(e^(2)))`

Explanation

To solve the problem, we need to determine how many electrons are missing from each ion given the force of repulsion between them. Let's break down the solution step by step.

### Step 1: Understand the Force Between Ions
The force of repulsion \( F \) between two positive ions, each carrying a charge \( q \), separated by a distance \( d \) can be expressed using Coulomb's law:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{q^2}{d^2} \]
where \( \epsilon_0 \) is the permittivity of free space.

### Step 2: Relate Charge to Missing Electrons
If \( n \) electrons are missing from each ion, the charge \( q \) of each ion can be expressed in terms of the charge of an electron \( e \):
\[ q = n \cdot e \]
where \( e \) is the charge of a single electron (approximately \( 1.6 \times 10^{-19} \) coulombs).

### Step 3: Substitute Charge into the Force Equation
Substituting \( q = n \cdot e \) into the force equation gives:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{(n \cdot e)^2}{d^2} \]
This simplifies to:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{n^2 \cdot e^2}{d^2} \]

### Step 4: Rearranging to Find \( n^2 \)
Rearranging the equation to solve for \( n^2 \):
\[ n^2 = \frac{4 \pi \epsilon_0 F d^2}{e^2} \]

### Step 5: Taking the Square Root
To find \( n \), we take the square root of both sides:
\[ n = \sqrt{\frac{4 \pi \epsilon_0 F d^2}{e^2}} \]

### Conclusion
Thus, the number of electrons missing from each ion is given by:
\[ n = \sqrt{\frac{4 \pi \epsilon_0 F d^2}{e^2}} \]