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: Concept Definition:
The Stefan-Boltzmann law quantifies the rate at which an object emits thermal energy. Total radiated energy is calculated by multiplying the rate of energy emission (power) by the duration.Step 2: Governing Equation:
For a non-ideal black body, the power (P) radiated is given by:
\[ P = e \sigma A T^4 \]where \(e\) is emissivity, \(\sigma\) is the Stefan-Boltzmann constant, A is the surface area, and T is the absolute temperature.
The total radiated energy (E) over a time interval (t) is:
\[ E = P \times t \]Step 3: Calculation Procedure:
1. Input Parameters:
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 interval, \( t = 1 \, \text{minute} = 60 \, \text{s} \). 2. Power Calculation:
\[ P = (0.85) \times (5.7 \times 10^{-8}) \times (4 \times 10^{-5}) \times (2000)^4 \]\[ P = (0.85) \times (5.7 \times 10^{-8}) \times (4 \times 10^{-5}) \times (16 \times 10^{12}) \]\[ P = (0.85 \times 5.7 \times 4 \times 16) \times (10^{-8} \times 10^{-5} \times 10^{12}) \]\[ P = 310.08 \times 10^{-1} = 31.008 \, \text{W (J/s)} \]3. Total Energy Calculation (1 minute):
\[ E = P \times t = 31.008 \, \text{J/s} \times 60 \, \text{s} = 1860.48 \, \text{J} \]A discrepancy is noted between the computed result and the provided options. Re-examination of options reveals \( 27.36 \times 60 = 1641.6 \). This suggests the intended power value was 27.36 W, possibly due to a variation in the Stefan's constant used (approximately \( 5.0 \times 10^{-8} \)). Assuming the 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 value corresponds precisely to option (D), indicating it is the likely intended answer given the multiple-choice format. Step 4: Conclusion:
Based on the assumption of a typo in the given constants leading to an intended power of 27.36 W, the total energy radiated in one minute is 1641.6 J.