First slide

Radiation

Question

The power radiated by a black body is P and it radiates maximum energy at wavelength, $λ_{0}$ . If the temperature of the black body is now changed so that it radiates maximum energy at wavelength $\frac{3}{4} λ_{0}$ , the power radiated by it becomes nP. The value of n is

Moderate

A

$\frac{81}{256}$
B

$\frac{3}{4}$
C

$\frac{256}{81}$
D

$\frac{4}{3}$

Solution

P = ${σAT}^{4} \Rightarrow P directly proportional to T^{4}$
According to Wein's law $T is directly proportional \frac{1}{λ_{m}}$
$\Rightarrow$ P $directly proporitonal to {(\frac{1}{λ_{m}})}^{4} \Rightarrow \frac{P_{2}}{P_{1}} = {(\frac{λ_{m_{1}}}{λ_{m_{2}}})}^{4}$
$\Rightarrow \frac{P_{2}}{P_{1}} = {(\frac{λ_{0}}{\frac{3}{4} λ_{o}})}^{4}$
$\frac{nP}{P} = \frac{256}{81} \Rightarrow n = \frac{256}{81}$

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