Maximum energy stored in a capacitor consisting dielectric constant of dielectric strength `U_(d)` and breakdown voltage `V= U_(d)d` is given by

Options

A. `(1)/(2) epsi AdU_(d)`
B. `(1)/(2) epsi Ad U_(d)^(2)`
C. `epsi Ad U_(d)^(2)`
D. `epsi A d U_(d)`

Explanation

To find the maximum energy stored in a capacitor with a dielectric, we can follow these steps:

### Step 1: Understand the Formula for Energy Stored
The energy \( U \) 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: Identify the Capacitance with Dielectric
The capacitance \( C \) of a capacitor with a dielectric is given by:
\[ C = \frac{\varepsilon A}{d} \]
where:
- \( \varepsilon \) is the permittivity of the dielectric material,
- \( A \) is the area of the plates,
- \( d \) is the separation between the plates.

### Step 3: Relate Permittivity to Dielectric Constant
The permittivity \( \varepsilon \) can be expressed in terms of the dielectric constant \( k \) (or \( U_d \) in this case) as:
\[ \varepsilon = k \varepsilon_0 \]
where \( \varepsilon_0 \) is the permittivity of free space.

### Step 4: Substitute Capacitance into Energy Formula
Substituting the expression for capacitance into the energy formula gives:
\[ U = \frac{1}{2} \left( \frac{k \varepsilon_0 A}{d} \right) V^2 \]

### Step 5: Substitute Breakdown Voltage
The breakdown voltage \( V \) is given by:
\[ V = U_d d \]
Substituting this into the energy formula, we get:
\[ U = \frac{1}{2} \left( \frac{k \varepsilon_0 A}{d} \right) (U_d d)^2 \]

### Step 6: Simplify the Expression
Now, simplify the expression:
\[ U = \frac{1}{2} \left( \frac{k \varepsilon_0 A}{d} \right) (U_d^2 d^2) \]
This simplifies to:
\[ U = \frac{1}{2} k \varepsilon_0 A U_d^2 d \]

### Final Expression for Maximum Energy
Thus, the maximum energy stored in the capacitor is:
\[ U = \frac{1}{2} k \varepsilon_0 A U_d^2 d \]