A battery charges a parallel plate capacitor of thickness `(d)` so that an energy `[U_(0)]` is stored in the system. A slab of dielectric constant `(K)` and thickness `(d)` is then introduced between the plates of the capacitor. The new energy of the system is given by

Options

A. `K U_(0)`
B. `K^(2) U_(0)`
C. `U_(0)/K`
D. `U_(0)/K^(2)`

Answer

To solve the problem step by step, we need to analyze the situation before and after the introduction of the dielectric slab in the parallel plate capacitor.

### Step 1: Understand the Initial Conditions
Initially, we have a parallel plate capacitor with:
- Thickness = \(d\)
- Initial energy stored = \(U_0\)

The energy stored in a capacitor is given by the formula:
\[ U = \frac{1}{2} C V^2 \]
where \(C\) is the capacitance and \(V\) is the voltage across the capacitor.

### Step 2: Determine the Initial Capacitance
The capacitance \(C\) of a parallel plate capacitor is given by:
\[ C = \frac{\varepsilon_0 A}{d} \]
where:
- \(\varepsilon_0\) is the permittivity of free space
- \(A\) is the area of the plates
- \(d\) is the separation between the plates

### Step 3: Relate Initial Energy to Capacitance and Voltage
From the initial energy stored \(U_0\), we can express it as:
\[ U_0 = \frac{1}{2} C V^2 \]
This can be rearranged to find \(CV^2\):
\[ CV^2 = 2U_0 \]

### Step 4: Introduce the Dielectric Slab
When a dielectric slab with dielectric constant \(K\) and thickness \(d\) is introduced between the plates, the new capacitance \(C'\) becomes:
\[ C' = K \cdot C \]

### Step 5: Calculate the New Energy Stored
The new energy \(U'\) stored in the capacitor after the dielectric is introduced can be expressed as:
\[ U' = \frac{1}{2} C' V^2 \]
Substituting \(C'\) into the equation:
\[ U' = \frac{1}{2} (K \cdot C) V^2 \]

### Step 6: Substitute \(CV^2\) from Initial Conditions
Now, we can substitute \(CV^2\) from the earlier step:
\[ U' = \frac{1}{2} (K \cdot C) V^2 = K \cdot \frac{1}{2} C V^2 = K \cdot U_0 \]

### Final Result
Thus, the new energy of the system after introducing the dielectric slab is:
\[ U' = K \cdot U_0 \]