Calculate the energy radiated per minute from a filament of an incandescent lamp at 3,000 K if the surface area is `10^(-4) m^(2)` and its relative emitted is 0.425.

Explanation

To calculate the energy radiated per minute from the filament of an incandescent lamp at a temperature of 3000 K, with a surface area of \(10^{-4} \, m^2\) and a relative emissivity of 0.425, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula**:
The energy radiated per unit time (power) by a black body is given by Stefan-Boltzmann Law:
\[ P = \epsilon \sigma T^4 A \]
where:
- \(P\) = power (energy per unit time)
- \(\epsilon\) = emissivity (relative emitted)
- \(\sigma\) = Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \, W/m^2K^4\))
- \(T\) = absolute temperature in Kelvin
- \(A\) = surface area in \(m^2\)

2. **Calculate the Power**:
Substitute the values into the formula:
\[ P = 0.425 \times 5.67 \times 10^{-8} \times (3000)^4 \times (10^{-4}) \]

3. **Calculate \(T^4\)**:
First, calculate \(3000^4\):
\[ 3000^4 = 81 \times 10^{12} \]

4. **Substitute \(T^4\) into the Power Equation**:
Now substitute \(T^4\) back into the power equation:
\[ P = 0.425 \times 5.67 \times 10^{-8} \times (81 \times 10^{12}) \times (10^{-4}) \]

5. **Simplify the Expression**:
Combine the constants:
\[ P = 0.425 \times 5.67 \times 81 \times 10^{12 - 8 - 4} = 0.425 \times 5.67 \times 81 \times 10^{0} \]

6. **Calculate the Numerical Value**:
Calculate \(0.425 \times 5.67 \times 81\):
\[ 0.425 \times 5.67 \approx 2.41 \]
\[ 2.41 \times 81 \approx 195.21 \, W \]

7. **Convert Power to Energy per Minute**:
To find the energy radiated per minute, multiply the power by the time (1 minute = 60 seconds):
\[ E = P \times t = 195.21 \, W \times 60 \, s = 11712.6 \, J \]

8. **Final Result**:
The energy radiated per minute from the filament is approximately:
\[ E \approx 11712.6 \, J \]