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

Answer

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the Initial Energy Stored in the Capacitor
The initial energy stored in the capacitor, when it is charged by a battery, is given by the formula:
\[ U_0 = \frac{1}{2} C V^2 \]
where \( C \) is the capacitance of the capacitor and \( V \) is the voltage across it.

### Step 2: Determine the Capacitance of the Capacitor
For a parallel plate capacitor, the capacitance \( C \) is given by:
\[ C = \frac{\epsilon_0 A}{d} \]
where \( \epsilon_0 \) is the permittivity of free space, \( A \) is the area of the plates, and \( d \) is the separation between the plates.

### Step 3: Introduce the Dielectric Material
When a dielectric slab with dielectric constant \( K \) and thickness \( d \) is introduced between the plates of the capacitor, the capacitance changes. The new capacitance \( C' \) becomes:
\[ C' = \frac{K \epsilon_0 A}{d} \]
This is because the presence of the dielectric increases the capacitance by a factor of \( K \).

### Step 4: Calculate the New Energy Stored in the Capacitor
The new energy stored in the capacitor with the dielectric slab introduced is given by:
\[ U' = \frac{1}{2} C' V^2 \]
Substituting \( C' \) into the equation:
\[ U' = \frac{1}{2} \left( \frac{K \epsilon_0 A}{d} \right) V^2 \]

### Step 5: Relate the New Energy to the Old Energy
Now, we can express the new energy \( U' \) in terms of the old energy \( U_0 \):
\[ U' = K \left( \frac{1}{2} \frac{\epsilon_0 A}{d} V^2 \right) = K U_0 \]
Thus, the new energy stored in the capacitor after introducing the dielectric is:
\[ U' = K U_0 \]

### Final Answer
The new energy of the system after introducing the dielectric is:
\[ U' = K U_0 \]

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