Question

The power radiated by a black body is P and it radiates maximum energy around the wavelength ${\lambda }_0$. If the temperature of the black body is now changed so that it radiates maximum energy around wavelength $\frac{3{\lambda }_0}{4}$, the power radiated by it will increase by a factor of

• A. $\frac{4}{3}$
• B. $\frac{16}{9}$
• C. $\frac{64}{27}$
• D. $\frac{256}{81}$

Answer 1

The correct option is D $\frac{256}{81}$

${\lambda }_{m}T$ = constant (From Wien's Law)
$\frac{{\lambda }_{m_1}}{{\lambda }_{m_2}}=\frac{T_2}{T_1}$
$\frac{T_2}{T_1}=\frac{{\lambda }_0}{\frac{3}{4}{\lambda }_0}=\frac{4}{3}\Rightarrow T_2=\frac{4}{3}{T}_1$

From Stefan-Boltzman's Law
$E^{\prime }=\sigma A{T}_2^4=\sigma A{\left(\frac{4}{3}{T}_1\right)}^4=E\times {\left(\frac{4}{3}\right)}^4=\frac{256}{81}E$
$\frac{E^{\prime }}{E}=\frac{256}{81}$