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 a wavelength $\frac{3{\lambda }_0}{4}$, the power radiated by it will increase by a factor of

• A. $4/3$
• B. $16/9$
• C. $64/27$
• D. $256/81$

Answer 1

The correct option is D $256/81$

Let, $T_0=$ initial temperature of the black body.
Since, $P\propto T^4$
$\therefore$ Power radiated, $P_0=C.T_0^4$.
$(C=$ constant)
According to wein's law,
$\lambda T=b$ (constant)
$\therefore {\lambda }_0{T}_0=b$ (constant)
Let, $T=$New temperature of black body.
$\therefore {\lambda }_0{T}_0=\lambda T$
$\frac{3{\lambda }_0}{4}T={\lambda }_0{T}_0$ or $T=\frac{4T_0}{3}$
Power radiated
$P=C.T^4=\left(C{T}_0^4\right){\left(\frac{4}{3}\right)}^4=P_0\left(\frac{256}{81}\right)$