Two similar point charges `q_1 and q_2` are placed at a distance r apart in the air. The force between them is `F_1` . A dielectric slab of thickness `t(ltr)` and dielectric constant K is placed between the charges . Then the force between the same charge . Then the fore between the same charges is `F_2` . The ratio is

Options

A. 1
B. K
C. `[(r-t+tsqrtK)/r]^2`
D. `[r/(r-t+tsqrtK)]^2`

Answer

To solve the problem, we need to find the ratio of the forces \( F_1 \) and \( F_2 \) acting between two similar point charges \( q_1 \) and \( q_2 \) placed at a distance \( r \) apart in air, and then with a dielectric slab of thickness \( t \) and dielectric constant \( K \) placed between them.

### Step-by-Step Solution:

1. **Understanding the Force Between Charges**:
The electrostatic force \( F \) between two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) in air (or vacuum) is given by Coulomb's Law:
\[ F_1 = \frac{k \cdot q_1 \cdot q_2}{r^2} \]
where \( k \) is Coulomb's constant.

2. **Inserting the Dielectric**:
When a dielectric slab of thickness \( t \) and dielectric constant \( K \) is inserted between the two charges, the effective distance between the charges changes. The dielectric reduces the effective force between the charges.

3. **Calculating the Effective Distance**:
The effective distance \( r' \) when the dielectric is inserted can be calculated as follows:
- The distance occupied by the dielectric slab is \( t \).
- The remaining distance in air (vacuum) is \( r - t \).
- The effective distance due to the dielectric is given by:
\[ r' = (r - t) + \frac{t}{\sqrt{K}} \]
This accounts for the fact that the dielectric reduces the force.

4. **Calculating the New Force**:
The new force \( F_2 \) when the dielectric is present is:
\[ F_2 = \frac{k \cdot q_1 \cdot q_2}{(r')^2} \]

5. **Finding the Ratio of Forces**:
To find the ratio of the forces \( \frac{F_1}{F_2} \):
\[ \frac{F_1}{F_2} = \frac{\frac{k \cdot q_1 \cdot q_2}{r^2}}{\frac{k \cdot q_1 \cdot q_2}{(r')^2}} = \frac{(r')^2}{r^2} \]

6. **Substituting for \( r' \)**:
Substitute \( r' \) into the equation:
\[ \frac{F_1}{F_2} = \frac{\left((r - t) + \frac{t}{\sqrt{K}}\right)^2}{r^2} \]

7. **Final Expression**:
The final expression for the ratio of the forces is:
\[ \frac{F_1}{F_2} = \frac{(r - t + \frac{t}{\sqrt{K}})^2}{r^2} \]