Question

Two perfectly black bodies are at temperatures \( T \) and \( 2T \). The ratio between the wavelengths corresponding to maximum energy emission by the two black bodies is

• A. 2 : 1
• B. 1 : 2
• C. 2 : 3
• D. 3 : 2
• E. 1 : 4

Hint:

Remember that Wien's displacement law can be crucial for understanding how temperature influences the spectral distribution of black body radiation.

Answer 1

The Correct Option is A

To solve the problem, we use Wien's Displacement Law, which relates the temperature of a black body to the wavelength at which it emits the maximum energy. 
The law is given by: \[ \lambda_{{max}} T = b \]
where: 
- \(\lambda_{{max}}\) is the wavelength corresponding to maximum energy emission, 
- \(T\) is the absolute temperature of the black body, 
- \(b\) is Wien's constant (\(b \approx 2.898 \times 10^{-3} \, {m·K}\)). 
Step 1: Apply Wien's Displacement Law For the two black bodies: 1. For the first black body at temperature \(T\): \[ \lambda_1 T = b \implies \lambda_1 = \frac{b}{T} \] 
For the second black body at temperature \(2T\): \[ \lambda_2 (2T) = b \implies \lambda_2 = \frac{b}{2T} \] 
Step 2: Find the ratio of wavelengths The ratio of the wavelengths corresponding to maximum energy emission is: \[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{b}{T}}{\frac{b}{2T}} = \frac{b}{T} \cdot \frac{2T}{b} = 2 \] Thus, the ratio is: \[ \lambda_1 : \lambda_2 = 2 : 1 \] 
Final Answer: The ratio between the wavelengths corresponding to maximum energy emission by the two black bodies is: \[ \boxed{2 : 1} \]