Two charge spheres separated at a distance d exert a force F on each other. If they are immersed in a liquid of dielectric constant K=2, then the force (if all conditions are same) is

Options

A. `(F)/(2)`
B. `F`
C. `2F`
D. `4F`

Answer

To solve the problem step by step, we will use the formula for the electrostatic force between two charged spheres and consider the effect of the dielectric medium.

### Step-by-Step Solution:

1. **Understand the Initial Force**:
The force \( F \) between two charged spheres separated by a distance \( d \) is given by Coulomb's law:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{d^2} \]
where \( q_1 \) and \( q_2 \) are the charges on the spheres, and \( \epsilon_0 \) is the permittivity of free space.

**Hint**: Remember that the force is inversely proportional to the square of the distance between the charges.

2. **Introduce the Dielectric Constant**:
When the charged spheres are immersed in a liquid with a dielectric constant \( K \), the force \( F' \) between the charges changes. The new force can be expressed as:
\[ F' = \frac{1}{4 \pi \epsilon} \frac{q_1 q_2}{d^2} \]
where \( \epsilon = K \epsilon_0 \). Thus, we can write:
\[ F' = \frac{1}{K} \cdot \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{d^2} \]

**Hint**: The dielectric constant reduces the effective force between the charges.

3. **Relate the Forces**:
From the above equations, we can relate the original force \( F \) to the new force \( F' \):
\[ F' = \frac{F}{K} \]

**Hint**: This relationship shows how the force changes when a dielectric is introduced.

4. **Substitute the Given Dielectric Constant**:
Given that the dielectric constant \( K = 2 \), we can substitute this value into the equation:
\[ F' = \frac{F}{2} \]

**Hint**: Plugging in the value of \( K \) directly gives you the new force.

5. **Conclusion**:
Therefore, the new force \( F' \) when the spheres are immersed in the liquid is:
\[ F' = \frac{F}{2} \]

**Final Answer**: The force exerted between the two charged spheres when immersed in the liquid is \( \frac{F}{2} \).