A 1.0 cm thick plate of dielectric material is placed midway between two copper plates 3.0 cm apart. If the capacity before introducing the dielectric is `C_(1)` and after introducing dielectric `C_(2)`. (K is dielectric constant).

Options

A. `C_(1)//C_(2)=(3)/(2+1//k)`
B. `C_(1)//C_(2)=(1)/(3)(2+k)`
C. `C_(1)//C_(2)=(1)/(3)((2)/(k)+1)`
D. `C_(1)//C_(2)=(1)/(3)((1)/(k)+2)`

Answer

To solve the problem, we need to find the ratio of the capacitance before and after introducing the dielectric material between the plates. Let's denote the following:

- \( D \) = distance between the plates = 3.0 cm = 0.03 m
- \( T \) = thickness of the dielectric = 1.0 cm = 0.01 m
- \( K \) = dielectric constant of the material
- \( C_1 \) = capacitance before introducing the dielectric
- \( C_2 \) = capacitance after introducing the dielectric

### Step 1: Calculate \( C_1 \)

The capacitance \( C_1 \) of the parallel plate capacitor without the dielectric is given by the formula:

\[ C_1 = \frac{\epsilon_0 A}{D} \]

where:
- \( \epsilon_0 \) is the permittivity of free space,
- \( A \) is the area of the plates,
- \( D \) is the distance between the plates.

### Step 2: Calculate \( C_2 \)

When the dielectric is introduced, the effective distance between the plates becomes \( D - T \). The capacitance \( C_2 \) with the dielectric is given by:

\[ C_2 = \frac{\epsilon_0 A}{D - T} \cdot K \]

### Step 3: Find the ratio \( \frac{C_1}{C_2} \)

Now, we can find the ratio of the capacitances:

\[ \frac{C_1}{C_2} = \frac{\frac{\epsilon_0 A}{D}}{\frac{\epsilon_0 A}{D - T} \cdot K} \]

This simplifies to:

\[ \frac{C_1}{C_2} = \frac{D - T}{D} \cdot \frac{1}{K} \]

### Step 4: Substitute the values

Substituting \( D = 0.03 \) m and \( T = 0.01 \) m into the equation:

\[ \frac{C_1}{C_2} = \frac{0.03 - 0.01}{0.03} \cdot \frac{1}{K} \]

This simplifies to:

\[ \frac{C_1}{C_2} = \frac{0.02}{0.03} \cdot \frac{1}{K} = \frac{2}{3} \cdot \frac{1}{K} \]

### Step 5: Final expression

Thus, the final expression for the ratio of capacitances is:

\[ \frac{C_1}{C_2} = \frac{2}{3K} \]