Question

According to Wien's displacement law, how is the surface temperature of a star related to its maximum intensity wavelength?

• A. $\lambda_{\text{max}} \propto T$
• B. $\lambda_{\text{max}} \propto \frac{1}{T}$
• C. $\lambda_{\text{max}} \propto T^2$
• D. $\lambda_{\text{max}} \propto \frac{1}{T^2}$

Hint:

Remember: Hotter object → shorter wavelength ($\lambda_{\text{max}} \downarrow$).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Wien's displacement law is a principle of black-body radiation.
It states that the black-body radiation curve for different temperatures peaks at a specific wavelength, and this peak wavelength is inversely proportional to the absolute temperature of the body.
Step 2: Key Formula or Approach:
The mathematical formulation of Wien's displacement law is:
\[ \lambda_{\max} \cdot T = b \]
where \(\lambda_{\max}\) is the wavelength at which the emission intensity is maximized, \(T\) is the absolute surface temperature (in Kelvin), and \(b\) is Wien's displacement constant (\(\approx 2.898 \times 10^{-3} \text{ m}\cdot\text{K}\)).
Step 3: Detailed Explanation:
Rearranging the formula to isolate \(\lambda_{\max}\):
\[ \lambda_{\max} = \frac{b}{T} \]
This explicitly demonstrates an inverse relationship between the maximum intensity wavelength and the temperature.
Therefore, \(\lambda_{\max} \propto \frac{1}{T}\).
Step 4: Final Answer:
The surface temperature is related to the maximum intensity wavelength by \(\lambda_{\max} \propto \frac{1}{T}\).