The potential enery of a charged parallel plate capacitor is `U_(0)`. If a slab of dielectric constant K is inserted between the plates, then the new potential energy will be

Options

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

Answer

To solve the problem, we need to determine the new potential energy of a charged parallel plate capacitor after inserting a dielectric slab with dielectric constant \( K \).

### Step-by-Step Solution:

1. **Initial Potential Energy**:
The potential energy \( U_0 \) of a charged parallel plate capacitor is given by the formula:
\[ U_0 = \frac{1}{2} C V^2 \]
where \( C \) is the capacitance and \( V \) is the voltage across the plates.

2. **Capacitance with Dielectric**:
When a dielectric slab with dielectric constant \( K \) is inserted, the capacitance of the capacitor increases. The new capacitance \( C' \) can be expressed as:
\[ C' = K C \]
This means the capacitance becomes \( K \) times the original capacitance.

3. **New Potential Energy**:
The new potential energy \( U \) of the capacitor with the dielectric inserted can be calculated using the new capacitance:
\[ U = \frac{1}{2} C' V'^2 \]
However, we need to consider how the voltage \( V' \) changes when the dielectric is inserted. If the capacitor is isolated (not connected to a battery), the charge \( Q \) remains constant. The relationship between charge, capacitance, and voltage is:
\[ Q = C V = C' V' \]
Since \( Q \) is constant, we can express \( V' \) in terms of \( V \):
\[ V' = \frac{Q}{C'} = \frac{Q}{K C} = \frac{V}{K} \]

4. **Substituting into the Potential Energy Formula**:
Now substituting \( C' \) and \( V' \) into the potential energy formula:
\[ U = \frac{1}{2} (K C) \left(\frac{V}{K}\right)^2 \]
Simplifying this gives:
\[ U = \frac{1}{2} K C \frac{V^2}{K^2} = \frac{1}{2} \frac{C V^2}{K} = \frac{U_0}{K} \]

5. **Final Result**:
Therefore, the new potential energy \( U \) after inserting the dielectric is:
\[ U = \frac{U_0}{K} \]

### Summary:
The new potential energy of the charged parallel plate capacitor after inserting a dielectric slab with dielectric constant \( K \) is \( \frac{U_0}{K} \).