Question

Two spheres $A$ and $B$ having radii $3\text{cm}$ and $5\text{cm}$ respectively are coated with carbon black on their outer surface. The wavelengths of maximum intensity of emission of radiation are $300\text{nm}$ and $500\text{nm}$ respectively. The respective powers radiated by them are in the ratio of :

• A. $\sqrt{\frac{5}{3}}$
• B. $\frac{5}{3}$
• C. ${\left(\frac{5}{3}\right)}^2$
• D. ${\left(\frac{5}{3}\right)}^4$

Answer 1

The correct option is C ${\left(\frac{5}{3}\right)}^2$

The bodies behave as black bodies.

Using Wien's displacement law :
${\lambda }_{m}T=b\Rightarrow T=\frac{b}{{\lambda }_m}...\left(i\right)$

Using Stefan's law, power radiated by a black body at temperature $T$
$P=\sigma A{T}^4...\left(i\right)$

Using (i) and (ii)
$P=\sigma A{\left(\frac{b}{{\lambda }_m}\right)}^4=\frac{\sigma b^{4}A}{{\lambda }_m^4}$

Area of sphere $=4\pi r^2$
$\Rightarrow P=\frac{\sigma b^4\left(4\pi r^2\right)}{{\lambda }_m^4}$
$\Rightarrow P\propto \frac{r^2}{{\lambda }_m^4}$
$\Rightarrow \frac{P_1}{P_2}=\frac{r_1^2}{r_2^2}\frac{({\lambda }_{m2}{)}^4}{({\lambda }_{m1}{)}^4}$
$\Rightarrow \frac{P_1}{P_2}={\left(\frac{3}{5}\right)}^2{\left(\frac{500}{300}\right)}^4$
$\Rightarrow \frac{P_1}{P_2}={\left(\frac{5}{3}\right)}^2$
is the ratio of powers radiated.

Thus, option (c) is the correct answer.