Question

A black body is at a temperature of $5760 \,K$. The energy of radiation emitted by the body at wavelength $250\,nm$ is $U_1$, at wavelength $500\, nm$ is $U_2$ and that at $1000\,nm$ is $U_3$. Wien's constant, $b = 2.88 \times 10^6 \, nm\,K$. Which of the following is correct ?

• A. $U_3 = 0$
• B. $U_1 > U_2$
• C. $U_2 > U_1$
• D. $U_1 = 0$

Answer 1

The Correct Option is C

 To solve this problem, we need to understand the concept of black body radiation and apply Wien's law. Wien's displacement law is given by the formula:

\[\lambda_{\text{max}} \cdot T = b\]

where \(\lambda_{\text{max}}\) is the wavelength corresponding to the maximum energy emitted, \(T\) is the absolute temperature of the black body, and \(b\) is Wien's constant.

Given:

• Temperature, \(T = 5760\,K\)
• Wien's constant, \(b = 2.88 \times 10^6 \, nm\,K\)

First, calculate \(\lambda_{\text{max}}\):

\[\lambda_{\text{max}} = \frac{b}{T} = \frac{2.88 \times 10^6 \, nm\,K}{5760\,K} = 500 \, nm\]

This means the black body emits maximum energy at a wavelength of \(500 \, nm\).

Now let's analyze the given wavelengths:

• \(250\, nm\)
• \(500\, nm\)
• \(1000\, nm\)

According to the concept of black body radiation, the energy emitted is highest around the peak wavelength and decreases as we move away from this peak.

Comparing the wavelengths:

• \(250\, nm\) is shorter and hence away from \(500 \, nm\)
• \(500\, nm\) is exactly at the peak
• \(1000\, nm\) is much longer and farther from \(500 \, nm\)

From this, we know:

• The energy at \(500\, nm\) (\(U_2\)) is the highest because it corresponds to peak emission.
• The energy at \(250\, nm\) (\(U_1\)) is less than at \(500\, nm\).
• The energy at \(1000\, nm\) (\(U_3\)) is significantly less compared to \(500\, nm\), and it is possible for \(U_3\) to be zero as it is far from the peak wavelength.

Given these analyses, we can conclude:

\(U_2 > U_1\).

This matches the correct option provided.