Two point charges placed at a certain distance r in air exert a force F on each other. Then the distance r at which these charges will exert the same force in a medium of dielectric constnat K is given by

Options

A. r
B. `r/K`
C. `r/(sqrt(K))`
D. `rsqrt(K)`

Answer

To solve the problem of finding the new distance \( r' \) at which two point charges exert the same force \( F \) in a medium with dielectric constant \( K \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Force Between Charges in Air**:
The electrostatic force \( F \) between two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) in air is given by Coulomb's law:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2} \]
For simplicity, we can assume \( q_1 = q_2 = q \):
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{q^2}{r^2} \]

2. **Force in a Medium with Dielectric Constant \( K \)**:
When the charges are placed in a medium with dielectric constant \( K \), the force \( F' \) between them is modified by the dielectric constant:
\[ F' = \frac{1}{4 \pi \epsilon_0 K} \frac{q^2}{(r')^2} \]

3. **Setting the Forces Equal**:
We want the force in the dielectric medium \( F' \) to be equal to the force in air \( F \):
\[ F = F' \]
Substituting the expressions for \( F \) and \( F' \):
\[ \frac{1}{4 \pi \epsilon_0} \frac{q^2}{r^2} = \frac{1}{4 \pi \epsilon_0 K} \frac{q^2}{(r')^2} \]

4. **Canceling Common Terms**:
Since \( \frac{1}{4 \pi \epsilon_0} \) and \( q^2 \) are common on both sides, we can cancel them:
\[ \frac{1}{r^2} = \frac{1}{K} \cdot \frac{1}{(r')^2} \]

5. **Rearranging the Equation**:
Rearranging the equation gives:
\[ (r')^2 = K \cdot r^2 \]

6. **Taking the Square Root**:
Taking the square root of both sides, we find:
\[ r' = r \sqrt{K} \]

7. **Final Expression**:
Since we want the distance \( r' \) at which the force remains the same, we can express it as:
\[ r' = \frac{r}{\sqrt{K}} \]

### Conclusion:
Thus, the distance \( r' \) at which the charges will exert the same force in a medium of dielectric constant \( K \) is given by:
\[ r' = \frac{r}{\sqrt{K}} \]