Question

The energy radiated per minute from the filament of an incandescent lamp at 2000 K, surface area \( 4 \times 10^{-5} \) m² and relative emittance is 0.85, will be (Given Stefan's constant \( \sigma = 5.7 \times 10^{-8} \) Jm⁻²s⁻¹K⁻⁴)

• A. 16.416 J
• B. 27.36 J
• C. 0.456 J
• D. 1641.6 J

Hint:

In exam questions with numerical calculations, if your result doesn't match any option, first double-check your own math. If it's correct, look for relationships between the options. Here, one option is 60 times another, a strong hint that they represent power (W or J/s) and energy per minute (J), respectively. This can guide you to the intended answer even if the problem data is slightly off.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The rate at which an object radiates thermal energy is described by the Stefan-Boltzmann law. The total energy radiated over a period of time is the power (rate of energy) multiplied by the time.
Step 2: Key Formula or Approach:
The Stefan-Boltzmann law for the power (P) radiated by a real object (not a perfect black body) is:
\[ P = e \sigma A T^4 \] where \(e\) is the emissivity (relative emittance), \(\sigma\) is the Stefan-Boltzmann constant, A is the surface area, and T is the absolute temperature.
The total energy (E) radiated in a time interval (t) is:
\[ E = P \times t \] Step 3: Detailed Explanation:
1. Identify the given values:
Emissivity, \( e = 0.85 \).
Stefan's constant, \( \sigma = 5.7 \times 10^{-8} \, \text{Jm}^{-2}\text{s}^{-1}\text{K}^{-4} \).
Surface area, \( A = 4 \times 10^{-5} \, \text{m}^2 \).
Temperature, \( T = 2000 \, \text{K} \).
Time, \( t = 1 \, \text{minute} = 60 \, \text{s} \).
2. Calculate the radiated power (P):
\[ P = (0.85) \times (5.7 \times 10^{-8}) \times (4 \times 10^{-5}) \times (2000)^4 \] \[ P = (0.85) \times (5.7 \times 10^{-8}) \times (4 \times 10^{-5}) \times (16 \times 10^{12}) \] \[ P = (0.85 \times 5.7 \times 4 \times 16) \times (10^{-8} \times 10^{-5} \times 10^{12}) \] \[ P = 310.08 \times 10^{-1} = 31.008 \, \text{W (J/s)} \] 3. Calculate the total energy (E) in one minute:
\[ E = P \times t = 31.008 \, \text{J/s} \times 60 \, \text{s} = 1860.48 \, \text{J} \] There seems to be a discrepancy between the calculated answer and the options provided. Let's re-examine the options. Note that \( 27.36 \times 60 = 1641.6 \). This suggests that the intended power calculation should have resulted in 27.36 W. This would happen if a slightly different value for Stefan's constant was used (approx \( 5.0 \times 10^{-8} \)). Assuming there is a typo in the provided constant and the intended power is 27.36 W:
Assumed Power, \( P = 27.36 \, \text{W} \).
Energy in 60 seconds, \( E = 27.36 \times 60 = 1641.6 \, \text{J} \). This matches option (D) exactly. Given the structure of the options, this is the most likely intended answer.
Step 4: Final Answer:
Assuming the intended power output is 27.36 W due to a likely typo in the constants provided, the total energy radiated in one minute is 1641.6 J.