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 epai_(0)Fd^(2))/(e)`
B. `sqrt((4 pi epsi_(0)Fe^(2))/(d^(2)))`
C. `sqrt((4 pi epsi_(0)Fd^(2))/(e^(2)))`
D. `(4 pi epsi_(0)Fd^(2))/(e^(2))`

Answer

To solve the problem of finding the number of electrons missing from each positive ion, we can follow these steps:

### Step 1: Understand the Force of Repulsion
The force of repulsion \( F \) between two positive charges \( q \) separated by a distance \( d \) is given by Coulomb's Law:
\[ F = \frac{k \cdot q_1 \cdot q_2}{d^2} \]
For two identical charges \( q \), this simplifies to:
\[ F = \frac{k \cdot q^2}{d^2} \]

### Step 2: Substitute the Value of \( k \)
The constant \( k \) can be expressed in terms of the permittivity of free space \( \epsilon_0 \):
\[ k = \frac{1}{4 \pi \epsilon_0} \]
Substituting this into the equation for \( F \):
\[ F = \frac{1}{4 \pi \epsilon_0} \cdot \frac{q^2}{d^2} \]

### Step 3: Relate Charge \( q \) to Number of Electrons
Let’s denote the number of electrons missing from each ion as \( n \). The charge \( q \) of each ion can be expressed as:
\[ q = n \cdot e \]
where \( e \) is the charge of a single electron.

### Step 4: Substitute \( q \) in 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 5: Rearranging for \( n \)
To find \( n \), we can rearrange the equation:
\[ n^2 = \frac{4 \pi \epsilon_0 \cdot F \cdot d^2}{e^2} \]
Taking the square root gives:
\[ n = \sqrt{\frac{4 \pi \epsilon_0 \cdot F \cdot d^2}{e^2}} \]

### Step 6: Conclusion
Thus, the number of electrons missing from each ion is:
\[ n = \sqrt{\frac{4 \pi \epsilon_0 \cdot F \cdot d^2}{e^2}} \]