Question

A parallel plate capacitor with air medium between the plates has a capacitance of $10 \mu\text{F}$. The area of capacitor is divided into two equal halves and filled with two media (as shown in figure) having dielectric constant $K_1 = 2$ and $K_2 = 4$. The capacitance of the system will be

• A. 10 \mu\text{F}
• B. 20 \mu\text{F}
• C. 30 \mu\text{F}
• D. 40 \mu\text{F}

Hint:

If the dielectric interface is perpendicular to the plates (area is divided), the parts are in parallel. If it is parallel to the plates (distance is divided), the parts are in series.

Answer 1

The Correct Option is C

Step 1: Identify the initial capacitance with air ($K=1$). \[ C_0 = \frac{\epsilon_0 A}{d} = 10 \mu\text{F} \]
Step 2: When the area is divided into two halves $A/2$, the configuration acts as two capacitors in parallel.
Step 3: Calculate the capacitance of the first half with dielectric $K_1 = 2$. \[ C_1 = \frac{K_1 \epsilon_0 (A/2)}{d} = \frac{K_1}{2} C_0 = \frac{2}{2} \times 10 = 10 \mu\text{F} \]
Step 4: Calculate the capacitance of the second half with dielectric $K_2 = 4$. \[ C_2 = \frac{K_2 \epsilon_0 (A/2)}{d} = \frac{K_2}{2} C_0 = \frac{4}{2} \times 10 = 20 \mu\text{F} \]
Step 5: Since they are in parallel, the total capacitance $C$ is the sum of the individual capacitances. \[ C = C_1 + C_2 = 10 + 20 = 30 \mu\text{F} \]