Question

A parallel plate air capacitor is charged upto $100\ \text{V}$. A plate $2\ \text{mm}$ thick is inserted between the plates. Then to maintain the same potential difference, the distance between the plates is increased by $1.6\ \text{mm}$. The dielectric constant of the thick plate is

• A. $4$
• B. $5$
• C. $2$
• D. $3$

Hint:

The effective "air equivalent" reduction in distance caused by inserting a dielectric slab is always $\Delta d = t(1 - \frac{1}{k})$. If the plates must be moved by $x$ to compensate, then $x$ must exactly equal this reduction: $x = t(1 - \frac{1}{k})$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
A dielectric slab is inserted into a capacitor, which inherently increases its capacitance. To restore the capacitance to its original value (thereby maintaining the same potential difference for the isolated charge), the plates are pulled further apart. We need to find the dielectric constant $k$.

Step 2: Key Formula or Approach:
The initial capacitance with air is $C = \frac{\epsilon_0 A}{d}$.
When a dielectric slab of thickness $t$ is inserted, the new capacitance is $C' = \frac{\epsilon_0 A}{d - t + \frac{t}{k}}$.
Since the distance is increased by $x$ to restore the original capacitance, the effective new distance is $d + x$. Thus, $C_{new} = \frac{\epsilon_0 A}{(d + x) - t + \frac{t}{k}}$. We equate this to the original $C$.

Step 3: Detailed Explanation:
Given:
Thickness of the slab, $t = 2\ \text{mm}$
Increase in distance, $x = 1.6\ \text{mm}$
Set the new capacitance equal to the original capacitance:
$$\frac{\epsilon_0 A}{d} = \frac{\epsilon_0 A}{(d + x) - t + \frac{t}{k}}$$
Cancel $\epsilon_0 A$ from both sides and equate the denominators:
$$d = d + x - t + \frac{t}{k}$$
Subtract $d$ from both sides:
$$0 = x - t + \frac{t}{k}$$
Rearrange to solve for $\frac{t}{k}$:
$$t - x = \frac{t}{k}$$
Substitute the given values ($t = 2$, $x = 1.6$):
$$2 - 1.6 = \frac{2}{k}$$
$$0.4 = \frac{2}{k}$$
Solve for $k$:
$$k = \frac{2}{0.4} = \frac{20}{4} = 5$$

Step 4: Final Answer:
The dielectric constant is $5$, matching option (B).