Two charges exert a force of 10 N on each other when separated by a distance 0.2 m in air. When they are placed in another medium of dielectric constant `K = 4`, and separated by distance R. they exert same force. The distance R equats to

Options

A. 2m
B. 1m
C. 0.2m
D. 0.1m

Answer

To solve the problem step by step, we will use Coulomb's law and the concept of dielectric constant.

### Step 1: Understand the initial conditions
We know that two charges exert a force of 10 N on each other when separated by a distance of 0.2 m in air. The formula for the force between two point charges is given by Coulomb's law:

\[ F = \frac{k \cdot q_1 \cdot q_2}{r^2} \]

Where:
- \( F \) is the force between the charges,
- \( k \) is the Coulomb's constant,
- \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
- \( r \) is the distance between the charges.

### Step 2: Set up the equation for the initial scenario
Given:
- \( F = 10 \, \text{N} \)
- \( r = 0.2 \, \text{m} \)

Substituting these values into Coulomb's law:

\[ 10 = \frac{k \cdot q_1 \cdot q_2}{(0.2)^2} \]

This simplifies to:

\[ 10 = \frac{k \cdot q_1 \cdot q_2}{0.04} \]

### Step 3: Rearrange to find \( k \cdot q_1 \cdot q_2 \)
Multiplying both sides by \( 0.04 \):

\[ k \cdot q_1 \cdot q_2 = 10 \cdot 0.04 = 0.4 \quad \text{(Equation 1)} \]

### Step 4: Consider the new conditions in the medium with dielectric constant
Now, the charges are placed in a medium with a dielectric constant \( K = 4 \) and separated by a distance \( R \). The force in the medium can be expressed as:

\[ F' = \frac{k \cdot q_1 \cdot q_2}{K \cdot R^2} \]

Since the force remains the same (\( F' = F = 10 \, \text{N} \)), we can set up the equation:

\[ 10 = \frac{k \cdot q_1 \cdot q_2}{4 \cdot R^2} \]

### Step 5: Substitute \( k \cdot q_1 \cdot q_2 \) from Equation 1
From Equation 1, we know \( k \cdot q_1 \cdot q_2 = 0.4 \). Substituting this into the new force equation:

\[ 10 = \frac{0.4}{4 \cdot R^2} \]

### Step 6: Solve for \( R^2 \)
Rearranging gives:

\[ 10 \cdot 4 \cdot R^2 = 0.4 \]

\[ 40 \cdot R^2 = 0.4 \]

Dividing both sides by 40:

\[ R^2 = \frac{0.4}{40} = 0.01 \]

### Step 7: Find \( R \)
Taking the square root of both sides:

\[ R = \sqrt{0.01} = 0.1 \, \text{m} \]

### Final Answer
The distance \( R \) is \( 0.1 \, \text{m} \).

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