According to the Stefan–Boltzmann (Stefan’s) law of thermal radiation, the total emissive power of an ideal black body is proportional to which function of its absolute temperature?
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AFirst power of temperature (T^1)
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BSecond power of temperature (T^2)
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CFourth power of temperature (T^4)
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DLogarithm of temperature (ln T)
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EInverse of temperature (1/T)
Answer
Correct Answer: Fourth power of temperature (T^4)
Explanation
Introduction / Context:Radiative heat transfer from ideal black bodies follows a fundamental law that connects temperature to radiant energy emission. Recognizing the T^4 dependence is essential for high-temperature furnace design, spacecraft thermal control, and heat loss estimates in power equipment.
Given Data / Assumptions:
- Black body is an ideal emitter with emissivity ε = 1.
- Total emissive power E_b sums radiation over all wavelengths.
Concept / Approach:Stefan–Boltzmann law states E_b = σ * T^4, where σ is the Stefan–Boltzmann constant. For real surfaces, E = ε * σ * T^4 with 0 < ε ≤ 1. The key takeaway is the fourth-power dependence on absolute temperature (kelvin). Small temperature increases can therefore create large changes in radiative heat transfer.
Step-by-Step Solution:
1) Write the ideal law: E_b = σ * T^4.2) Identify the proportionality: E_b ∝ T^4.3) Select the option corresponding to the fourth power.Verification / Alternative check:Dimensional and empirical verifications are standard in heat transfer texts and laboratory black-body furnaces; the T^4 trend is well established.
Why Other Options Are Wrong:
- T^1, T^2, ln T, or 1/T: Do not represent total emissive power for a black body.
Common Pitfalls:Confusing monochromatic laws (Planck’s law, Wien’s displacement) with total emissive power; forgetting to use absolute temperature in kelvin.
Final Answer:Fourth power of temperature (T^4)