First slide

Radiation

Question

The energy spectrum of a black body exihibits maximum around a wavelength $λ_{o}$ . The temperature of the black body is now changed such that the energy is maximum around a wavelength $3 λ_{o} / 4$ . The power radiated by the black body will now increase by a factor of

Moderate

A

256/81
B

64/27
C

16/9
D

4/3

Solution

According to Wien's displacement law,
$\begin{aligned} {(λ_{m})}_{1} T_{1} = {(λ_{m})}_{2} T_{2} \\ ∴ λ_{0} T_{1} = (3 λ_{0} / 4) T_{2} \end{aligned}$
or $\frac{T_{1}}{T_{2}} = (\frac{3}{4}) . . . (i)$
We know that $\frac{E_{1}}{E_{2}} = \frac{T_{1}^{4}}{T_{2}^{4}}$
$∴ \frac{E_{2}}{E_{1}} = {(\frac{T_{2}}{T_{1}})}^{4} . . . . (ii)$
From eqns. (i) and (ii), We get
$\frac{E_{2}}{E_{1}} = {(\frac{4}{3})}^{4} = \frac{256}{81}$

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