Question

The capacity of air filled parallel plate capacitor is $C_0$. One-half of the space between the plates is filled with a dielectric constant 'K' as shown in figure. The new capacity becomes $C_n$. The ratio $C_n$ to $C_0$ is ______.

• A. $\frac{K+1}{2}$
• B. $\frac{K+1}{3}$
• C. $\frac{K+1}{4}$
• D. $4(K+1)$

Hint:

Dielectric split vertically (Area divided) $\rightarrow$ Capacitors are in Parallel ($C_{eq} = C_1 + C_2$).
Dielectric split horizontally (Distance divided) $\rightarrow$ Capacitors are in Series ($1/C_{eq} = 1/C_1 + 1/C_2$).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
When the dielectric fills half the area (split vertically), the setup acts as two capacitors in parallel. If it fills half the thickness (split horizontally), they are in series. In the standard "shown in figure" case for this ratio, it's a parallel combination.

Step 2: Formula Application:
$C_0 = \frac{\epsilon_0 A}{d}$. For the new setup: $C_1 = \frac{\epsilon_0 (A/2)}{d}$ (air) and $C_2 = \frac{K \epsilon_0 (A/2)}{d}$ (dielectric).

Step 3: Explanation:
$C_n = C_1 + C_2 = \frac{\epsilon_0 A}{2d} + \frac{K \epsilon_0 A}{2d} = \frac{\epsilon_0 A}{2d}(1 + K)$. $C_n = \frac{C_0}{2}(K + 1)$. The ratio $\frac{C_n}{C_0} = \frac{K+1}{2}$.

Step 4: Final Answer:
The ratio is $\frac{K+1}{2}$.