A dielectric slab of length l, width b, thickness d and dielectric constant K fills the space inside a parallel plate capacitor. At t = 0, the slab begins to be pulled out slowly with speed y. At time t, the capacity of the capacitor is

Options

A. `(epsilon_0b)/d[Kl-(K-1)vt]`
B. `(epsilon_0b)/b[Kl+(K+1)vt]`
C. `(epsilon_0b)/b[(Kl+1)vt]`
D. `(epsilon_0b)/b[l+(K-1)vt]`

Answer

To solve the problem step by step, we need to analyze the situation of the parallel plate capacitor with a dielectric slab being pulled out.

### Step-by-Step Solution:

1. **Identify the Parameters**:
- Length of the capacitor plates: \( l \)
- Width of the capacitor plates: \( b \)
- Thickness of the dielectric slab: \( d \)
- Dielectric constant of the slab: \( K \)
- Speed at which the slab is pulled out: \( v \)
- Distance moved by the slab at time \( t \): \( x = vt \)

2. **Determine the Remaining Distance**:
- The remaining distance of the dielectric slab inside the capacitor at time \( t \) is:
\[ l - x = l - vt \]

3. **Capacitance Calculation**:
- The capacitor can be divided into two parts:
- \( C_1 \): The part without the dielectric (length \( x \))
- \( C_2 \): The part with the dielectric (length \( l - x \))
- The areas for these sections are:
- Area of \( C_1 \): \( A_1 = x \cdot b = vt \cdot b \)
- Area of \( C_2 \): \( A_2 = (l - x) \cdot b = (l - vt) \cdot b \)

4. **Capacitance Formulas**:
- The capacitance without the dielectric:
\[ C_1 = \frac{A_1 \epsilon_0}{d} = \frac{(vt \cdot b) \epsilon_0}{d} \]
- The capacitance with the dielectric:
\[ C_2 = \frac{A_2 K \epsilon_0}{d} = \frac{((l - vt) \cdot b) K \epsilon_0}{d} \]

5. **Total Capacitance**:
- Since \( C_1 \) and \( C_2 \) are in parallel, the total capacitance \( C \) is:
\[ C = C_1 + C_2 \]
- Substituting the values:
\[ C = \frac{(vt \cdot b) \epsilon_0}{d} + \frac{((l - vt) \cdot b) K \epsilon_0}{d} \]

6. **Simplifying the Expression**:
- Factor out common terms:
\[ C = \frac{b \epsilon_0}{d} \left( vt + K(l - vt) \right) \]
- Distributing \( K \):
\[ C = \frac{b \epsilon_0}{d} \left( K l + vt(1 - K) \right) \]

7. **Final Expression**:
- Therefore, the capacitance of the capacitor at time \( t \) is:
\[ C(t) = \frac{b \epsilon_0}{d} \left( Kl + vt(1 - K) \right) \]