Question

A parallel-plate capacitor of area $A$, plate separation d and capacitance $C$ is filled with four dielectric materials having dielectric constants $k_1, k_2, k_3 $ and $ k_4$ as shown in the figure below. If a single dielectric material is to be used to have the same capacitance $C$ in this capacitor, then its dielectric constant $k$ is given by

• A. $k = k _1 + k_2 + k_3 + 3k_4$
• B. $k = \frac{2}{3} (k_1 + k_2 + k_3) + 2k_4$
• C. $\frac{2}{k} = \frac{3}{k_1 + k_2 + k_3} + \frac{1}{k_4}$
• D. $\frac{1}{k} = \frac{1}{k_1 } + \frac{1}{k_2} + \frac{1}{k_3} + \frac{3}{2 k_4}$

Answer 1

The Correct Option is C

To solve this problem, we need to determine the equivalent dielectric constant $k$ for a capacitor filled with four different dielectric materials, each with its own dielectric constant $k_1, k_2, k_3, and $k_4$. Here's how we can approach this:

1. **Understanding the Configuration**: The capacitor is divided into regions, each filled with a different dielectric material. This affects the overall capacitance as a combination of capacitors in series and parallel, depending on the configuration mentioned in the problem.

2. **Formula for Capacitance**: The capacitance $C$ of a parallel-plate capacitor filled with a dielectric is given by:

$$ C = \frac{\varepsilon_0 \cdot k \cdot A}{d} $$

where $ \varepsilon_0 $ is the permittivity of free space, $A$ is the plate area, $d$ is the separation between the plates, and $k$ is the dielectric constant.

3. **Combination of Dielectrics**: In this problem, four dielectrics are combined. Depending on their configuration, equivalent capacitance formula may change:

• If placed serially, the inverse capacitance is the sum of inverse capacitances.
• If placed in parallel, the sum of capacitances is direct.

4. **Analysis of Options**: We need to find the equivalent dielectric constant $k$ such that the combined effect remains as specified capacitance $C$.

5. **Correct Answer**: Let's analyze each option provided if option can maintain same capacitance:

The correct formula for this configuration is:

$$\frac{2}{k} = \frac{3}{k_1 + k_2 + k_3} + \frac{1}{k_4}$$

This equation indicates a serial combination of dielectrics where particular arrangement results in segments in series and parallel might provide such combination further reduced to given format to describe the overall dielectric constant.

6. **Discussion**: This leads us to conclude that the dielectric materials are combined in both series and parallel to provide this effective dielectric constant.

Hence, the correct answer is:

$$\frac{2}{k} = \frac{3}{k_1 + k_2 + k_3} + \frac{1}{k_4}$$