To tackle this problem, we need to apply concepts from thermal radiation and the Stefan-Boltzmann law, as well as Wien's displacement law. Let's break it down step by step.

Understanding Thermal Emissivity

Thermal emissivity is a measure of how effectively a body emits thermal radiation compared to a perfect black body. In this case, body A has an emissivity of 0.01, while body B has an emissivity of 0.81. Since both bodies emit total radiant power at the same rate, we can use their emissivities to relate their temperatures.

Applying the Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the total power emitted per unit area of a black body is proportional to the fourth power of its absolute temperature. The formula is given by:

• P = εσT^4

Where:

• P = total power emitted
• ε = emissivity
• σ = Stefan-Boltzmann constant (approximately 5.67 × 10^-8 W/m²K⁴)
• T = absolute temperature in Kelvin

Since both bodies emit the same total power, we can set their equations equal to each other:

• ε_A * σ * T_A^4 = ε_B * σ * T_B^4

Substituting the values for emissivity and temperature for body A:

• 0.01 * σ * (5802)^4 = 0.81 * σ * T_B^4

We can cancel σ from both sides:

• 0.01 * (5802)^4 = 0.81 * T_B^4

Calculating the Temperature of Body B

Now, let's calculate the left side:

• (5802)^4 = 1.113 * 10^{15} K^4

Now substituting this value back into the equation:

• 0.01 * 1.113 * 10^{15} = 0.81 * T_B^4

This simplifies to:

• 1.113 * 10^{13} = 0.81 * T_B^4

Now, solving for T_B^4:

• T_B^4 = (1.113 * 10^{13}) / 0.81

Calculating this gives:

• T_B^4 ≈ 1.373 * 10^{13}

Taking the fourth root to find T_B:

• T_B ≈ (1.373 * 10^{13})^{1/4} ≈ 5860 K

Determining Wavelength λ_b

Next, we need to find the wavelength corresponding to the maximum spectral radiance for body B. We can use Wien's displacement law, which states:

• λ_max * T = b

Where b is Wien's displacement constant, approximately equal to 2898 μm·K. For body A, we can calculate λ_max:

• λ_max_A = b / T_A = 2898 μm·K / 5802 K ≈ 0.499 μm

Since the problem states that the wavelength for body B is shifted by 1 μm from body A, we can find λ_max_B:

• λ_max_B = λ_max_A + 1 μm ≈ 0.499 μm + 1 μm = 1.499 μm

Final Results

To summarize:

• Temperature of Body B (T_B): Approximately 5860 K
• Wavelength λ_b: Approximately 1.499 μm

This approach combines the principles of thermal radiation and allows us to derive the required values systematically. If you have any further questions or need clarification on any part, feel free to ask!