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

Answer

To solve the problem, we need to find the number of electrons missing from each ion given the force of repulsion between them. Let's break this down step by step.

### Step 1: Understanding 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:
\[ F = \frac{1}{4 \pi \epsilon_0} \cdot \frac{q^2}{d^2} \]
where \( \epsilon_0 \) is the permittivity of free space.

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

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

### Step 4: Rearranging the Equation to Solve for \( n^2 \)
From the equation above, we can rearrange it to find \( n^2 \):
\[ n^2 = \frac{4 \pi \epsilon_0 \cdot F \cdot d^2}{e^2} \]

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

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