Question

If two charges \( q_1 \) and \( q_2 \) are separated with distance 'd' and placed in a medium of dielectric constant K. What will be the equivalent distance between charges in air for the same electrostatic force?

• A. d√K
• B. K√d
• C. 1.5d√K
• D. 2d√K

Hint:

The electrostatic force in a medium is \( \frac{1}{K} \) times the force in vacuum for the same charges and distance. To maintain the same force, increase the distance in vacuum by \( \sqrt{K} \) times the distance in the medium.

Answer 1

The Correct Option is A

The electrostatic force between two charges in a medium with dielectric constant K is given by: \[ F_{medium} = \frac{1}{4\pi\epsilon_0 K} \frac{|q_1 q_2|}{d^2} \] where \( \epsilon_0 \) is the permittivity of free space. The electrostatic force between the same two charges in air (or vacuum, with K=1) at a distance \( d' \) is given by: \[ F_{air} = \frac{1}{4\pi\epsilon_0} \frac{|q_1 q_2|}{d'^2} \] For the forces to be equal, \( F_{medium} = F_{air} \): \[ \frac{1}{4\pi\epsilon_0 K} \frac{|q_1 q_2|}{d^2} = \frac{1}{4\pi\epsilon_0} \frac{|q_1 q_2|}{d'^2} \] \[ \frac{1}{K d^2} = \frac{1}{d'^2} \] \[ d'^2 = K d^2 \] \[ d' = \sqrt{K d^2} = d\sqrt{K} \] Thus, the equivalent distance in air is \( d\sqrt{K} \).

Answer 2

Step 1: Electrostatic force in a dielectric medium
The electrostatic force between two charges \( q_1 \) and \( q_2 \) separated by distance \( d \) in a medium of dielectric constant \( K \) is given by:
\[ F_{\text{medium}} = \frac{1}{4\pi\varepsilon_0 K} \cdot \frac{q_1 q_2}{d^2} \]

Step 2: Electrostatic force in air
Let the equivalent distance in air be \( d_{\text{eq}} \). Then the electrostatic force in air for the same charges is:
\[ F_{\text{air}} = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q_1 q_2}{d_{\text{eq}}^2} \]

Step 3: Equating the two forces
To find the equivalent distance for the same force:
\[ F_{\text{medium}} = F_{\text{air}} \Rightarrow \frac{1}{4\pi\varepsilon_0 K} \cdot \frac{q_1 q_2}{d^2} = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q_1 q_2}{d_{\text{eq}}^2} \]

Step 4: Simplify the equation
Cancel common terms:
\[ \frac{1}{K d^2} = \frac{1}{d_{\text{eq}}^2} \Rightarrow d_{\text{eq}}^2 = K d^2 \Rightarrow d_{\text{eq}} = d \sqrt{K} \]

Final Answer: \( d\sqrt{K} \)