Separation between the plates of a parallel plate capacitor is `d` and the area of each plates is `A`. When a slab of material of dielectric constant `k` and thickness `t(t lt d)` is introduced between the plates. Its capacitance becomes

Options

A. `(epsilon_(0)A)/(d+t(1-(1)/(K)))`
B. `(epsilon_(0)A)/(d+t(1+(1)/(K)))`
C. `(epsilon_(0)A)/(d-t(1-(1)/(K)))`
D. `(epsilon_(0)A)/(d-t(1+(1)/(K)))`

Answer

To find the capacitance of a parallel plate capacitor when a dielectric slab is introduced, we can follow these steps:

### Step 1: Understand the Configuration
We have a parallel plate capacitor with:
- Plate separation: \( d \)
- Area of each plate: \( A \)
- Dielectric slab thickness: \( t \) (where \( t < d \))
- Dielectric constant of the slab: \( k \)

### Step 2: Determine the Effective Capacitance
When the dielectric slab is introduced, the capacitor can be thought of as two capacitors in series:
1. The first capacitor has a dielectric slab of thickness \( t \) and dielectric constant \( k \).
2. The second capacitor has air (or vacuum) with thickness \( d - t \).

### Step 3: Calculate Capacitance of Each Section
1. **Capacitance of the dielectric section**:
\[ C_1 = \frac{k \epsilon_0 A}{t} \]
where \( \epsilon_0 \) is the permittivity of free space.

2. **Capacitance of the air section**:
\[ C_2 = \frac{\epsilon_0 A}{d - t} \]

### Step 4: Combine the Capacitances
Since \( C_1 \) and \( C_2 \) are in series, the total capacitance \( C \) is given by:
\[ \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} \]

Substituting the expressions for \( C_1 \) and \( C_2 \):
\[ \frac{1}{C} = \frac{t}{k \epsilon_0 A} + \frac{d - t}{\epsilon_0 A} \]

### Step 5: Simplify the Expression
Combine the fractions:
\[ \frac{1}{C} = \frac{t + (d - t)k}{k \epsilon_0 A} \]
\[ \frac{1}{C} = \frac{d k}{k \epsilon_0 A} \]

### Step 6: Solve for Capacitance
Taking the reciprocal gives:
\[ C = \frac{k \epsilon_0 A}{d k + (d - t)} \]

### Final Expression
Thus, the capacitance of the capacitor with the dielectric slab inserted is:
\[ C = \frac{A \epsilon_0}{\frac{d - t}{k} + t} \]