Question

The energy radiated by a black body at $2300\text{K}$ is found to have maximum intensity at wavelength $1260\text{nm}$, its emissive power being $8000{\text{W/m}}^2$. When the body is cooled to a temperature $T\text{K}$, the emissive power is found to decrease to $500{\text{W/m}}^2$. Find the wavelength at which intensity of emission is maximum at the temperature $T$.

• A. $1575\text{nm}$
• B. $2520\text{nm}$
• C. $1260\text{nm}$
• D. $5040\text{nm}$

Answer 1

The correct option is B $2520\text{nm}$

${\lambda }_{max}=1260\times {10}^{-9}\text{m}$ at $T_1=2300\text{K}$
Also, $T_2=T\text{K}$

We know that $E\propto T^4$
$\Rightarrow \frac{E_1}{E_2}={\left(\frac{T_1}{T_2}\right)}^4\Rightarrow T_2=T_1{\left[\frac{E_2}{E_1}\right]}^{1/4}$
$\Rightarrow T_2=2300\times {\left[\frac{500}{8000}\right]}^{1/4}$
$\Rightarrow T_2=2300\times \frac{1}{2}=1150\text{K}$
By Wein's law,
$T_1{\lambda }_1=T_2{\lambda }_2$
$\Rightarrow {\lambda }_2=\frac{T_1{\lambda }_1}{T_2}=\frac{2300\times 1260\times {10}^{-9}}{1150}$
$\Rightarrow {\lambda }_2=2520\text{nm}$