`N` fundamental charges each of charge `q` are to be distributed as two point charges separated by a fixed distance ,then the maximum to minimum force bears a ratio (N is even and greater than `2`)

Options

A. `(N-1)^(2)/(4N^(2))`
B. `(4N^(2))/((N-1))`
C. `(N^(2))/(4(N-1))`
D. `(2N^(2))/((N-1))`

Explanation

To solve the problem, we need to find the ratio of the maximum to minimum force between two point charges formed by distributing `N` fundamental charges, each of charge `q`, separated by a fixed distance `r`.

### Step-by-Step Solution:

1. **Understanding the Charges Distribution**:
- We have `N` fundamental charges, each of charge `q`.
- We can distribute these charges into two groups: one group with `n` charges and the other with `N - n` charges.

2. **Calculating the Force Between the Charges**:
- The force between two point charges is given by Coulomb's Law:
\[ F = k \frac{q_1 q_2}{r^2} \]
- Here, \( q_1 = n \cdot q \) and \( q_2 = (N - n) \cdot q \).
- Substituting these into the formula gives:
\[ F = k \frac{(n \cdot q)((N - n) \cdot q)}{r^2} = k \frac{n(N - n)q^2}{r^2} \]

3. **Finding Maximum and Minimum Forces**:
- The expression for force can be rewritten as:
\[ F = k \frac{q^2}{r^2} \cdot n(N - n) \]
- To maximize and minimize \( n(N - n) \), we recognize that this is a quadratic function in terms of \( n \).

4. **Using AM-GM Inequality**:
- The maximum value of \( n(N - n) \) occurs when \( n = \frac{N}{2} \) (since \( N \) is even).
- At this point:
\[ F_{\text{max}} = k \frac{q^2}{r^2} \cdot \left(\frac{N}{2}\right)\left(N - \frac{N}{2}\right) = k \frac{q^2}{r^2} \cdot \frac{N^2}{4} \]

5. **Finding the Minimum Force**:
- The minimum value occurs when \( n = 1 \):
\[ F_{\text{min}} = k \frac{q^2}{r^2} \cdot 1(N - 1) = k \frac{q^2}{r^2} \cdot (N - 1) \]

6. **Calculating the Ratio of Maximum to Minimum Force**:
- Now we find the ratio:
\[ \text{Ratio} = \frac{F_{\text{max}}}{F_{\text{min}}} = \frac{k \frac{q^2}{r^2} \cdot \frac{N^2}{4}}{k \frac{q^2}{r^2} \cdot (N - 1)} = \frac{\frac{N^2}{4}}{N - 1} \]
- Simplifying this gives:
\[ \text{Ratio} = \frac{N^2}{4(N - 1)} \]

### Final Answer:
The ratio of the maximum to minimum force is:
\[ \frac{N^2}{4(N - 1)} \]