A `450 muF` capacitor is charged to `295 V`. Then, a wire is connected between the plates. How many joule of thermal energy are produced as the capacitor discharges if all of the energy that was stored goes into heating the wire?

Explanation

To solve the problem of how much thermal energy is produced when a charged capacitor discharges, we can follow these steps:

### Step 1: Understand the energy stored in a capacitor
The energy (U) stored in a capacitor is given by the formula:
\[ U = \frac{1}{2} C V^2 \]
where:
- \( C \) is the capacitance in farads (F)
- \( V \) is the voltage in volts (V)

### Step 2: Convert the capacitance from microfarads to farads
The given capacitance is \( 450 \mu F \). To convert this to farads:
\[ C = 450 \mu F = 450 \times 10^{-6} F = 4.5 \times 10^{-4} F \]

### Step 3: Substitute the values into the energy formula
Now, substitute \( C \) and \( V \) into the energy formula:
\[ U = \frac{1}{2} \times (450 \times 10^{-6}) \times (295)^2 \]

### Step 4: Calculate \( V^2 \)
First, calculate \( V^2 \):
\[ 295^2 = 87025 \]

### Step 5: Substitute \( V^2 \) back into the energy formula
Now substitute \( V^2 \) back into the equation:
\[ U = \frac{1}{2} \times (450 \times 10^{-6}) \times 87025 \]

### Step 6: Calculate the energy stored
Calculate the energy:
\[ U = \frac{1}{2} \times 450 \times 10^{-6} \times 87025 \]
\[ U = 0.5 \times 450 \times 87025 \times 10^{-6} \]
\[ U = 225 \times 87025 \times 10^{-6} \]
\[ U = 19658.125 \times 10^{-6} \]
\[ U \approx 19.658125 \, J \]

### Step 7: Round the answer
The final answer can be rounded to:
\[ U \approx 19.6 \, J \]

### Final Answer
The thermal energy produced as the capacitor discharges is approximately **19.6 joules**.

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