Question

The spectral energy density u_T(λ) vs wavelength (λ) curve of a black body shows a peak at λ = λ_max. If the temperature of the black body is doubled, then

• A. the maximum of u_T(λ) shifts to λ_max/2
• B. the maximum of u_T(λ) shifts to 2λ_max
• C. the area under the curve becomes 16 times the original area
• D. the area under the curve becomes 8 times the original area

Answer 1

The Correct Option is A, B

The problem involves understanding the behavior of a black body and its spectral energy density, particularly how the peak wavelength of the black body radiation shifts when the temperature changes. This is governed by Wien's Displacement Law, which states:

\(\lambda_{\text{max}} T = b\) 

where \(\lambda_{\text{max}}\) is the wavelength corresponding to the maximum spectral energy density, \(T\) is the absolute temperature of the black body, and \(b\) is Wien's constant (approximately \(2.898 \times 10^{-3} \, \text{m} \cdot \text{K}\)).

Let's analyze the given statement:

1. We know from Wien's Law that if we double the temperature \(T\), that is:

\(T \rightarrow 2T\),

1. The product \(\lambda_{\text{max}} T = b\) must hold. Therefore,

\(\lambda_{\text{max,new}} \cdot 2T = b\)

1. This implies that

\(\lambda_{\text{max,new}} = \frac{\lambda_{\text{max}}}{2}\).

1. The conclusion is that when the temperature of the black body is doubled, the peak wavelength is halved. Therefore, the maximum of \(u_T(\lambda)\) shifts from \(\lambda_{\text{max}}\) to \(\frac{\lambda_{\text{max}}}{2}\).

Next, let's examine the behavior of the area under the curve:

The area under the spectral energy density curve is representative of the total energy emitted by the black body, which is given by the Stefan-Boltzmann Law:

\(E = \sigma T^4\)

Where \(\sigma\) is the Stefan-Boltzmann constant.

1. When the temperature is doubled (\(T \to 2T\)), the energy becomes:

\(E' = \sigma (2T)^4 = \sigma \cdot 16T^4 = 16E\)

1. Therefore, the area under the curve becomes 16 times the original area.

In conclusion, the correct choices are:

• The peak of \(u_T(\lambda)\) shifts to \(\frac{\lambda_{\text{max}}}{2}\).
• The area under the curve becomes 16 times the original area.

Thus, the correct answer is: "the maximum of \(u_T(\lambda)\) shifts to \(\lambda_{\text{max}}/2\)" and "the area under the curve becomes 16 times the original area".