Question

A black body has wavelength $\lambda_m$ corresponding to maximum energy at 2000 K. Its wavelength corresponding to maximum energy at 3000 K will be:

• A. $\frac{3}{2} \lambda_m$
• B. $\frac{2}{3} \lambda_m$
• C. $\frac{16}{81} \lambda_m$
• D. $\frac{81}{16} \lambda_m$

Answer 1

The Correct Option is B

The relation between the wavelength corresponding to maximum intensity of radiation at any temperature is given by Wien's displacement law. 
Wien's displacement law is given by 
\({{\lambda }_{m}}T=\text{cosntant}\) 
or \({{\lambda }_{1}}{{T}_{1}}={{\lambda }_{2}}{{T}_{2}}\)
 \(or{{\lambda }_{2}}={{\lambda }_{1}}\left( \frac{{{T}_{1}}}{{{T}_{2}}} \right)\) 
Here, \({{T}_{1}}=2000\,K,\,{{T}_{2}}=3000\,K,\,{{\lambda }_{1}}=\lambda\)
\(\therefore {{\lambda }_{2}}=\lambda \times \frac{2000}{3000}=\frac{2}{3}\,\lambda\)

So, the correct option is (B): \(\frac{2}{3} \lambda_m\)