Three discs, A , B and C , having radii 2m , 4m and 6m ,...

Three discs, $A$ , $B$ and $C$ , having radii $2m$ , $4m$ and $6m$ , respectively, are coated with carbon black on their outer surfaces. The wavelengths corresponding to maximum intensity are $300$ nm, $400$ nm and $500$ nm, respectively. The power radiated by them are ${Q}_{A}$ , ${Q}_{B}$ and ${Q}_{C}$ respectively. The correct statement, among the following, is

Options

A ${Q}_{A}$ is maximum
B ${Q}_{B}$ is maximum
C ${Q}_{C}$ is maximum
D ${Q}_{A} = {Q}_{B} = {Q}_{C}$

Correct answer

B. ${Q}_{B}$ is maximum

Step-by-step solution

\text{Q} ∝ {AT}^{4} \text{and} {λ}_{m} \text{T} = \text{Constant}

\text{Hence}

\text{Q} ∝ \dfrac{\text{A}}{{\left({λ}_{m}\right)}^{4}} \text{or} \text{Q} ∝ \dfrac{r^{2}}{{\left({λ}_{m}\right)}^{4}}

{Q}_{A}\text{:}{Q}_{B}\text{:}{Q}_{C} = \dfrac{\left(2\right)^{2}}{\left(3\right)^{4}}\text{:}\dfrac{\left(4\right)^{2}}{\left(4\right)^{4}}\text{:}\dfrac{\left(6\right)^{2}}{\left(5\right)^{4}}

= \dfrac{4}{8 1} \text{:}\dfrac{1}{1 6} \text{:} \dfrac{3 6}{6 2 5} = \text{0.05 : 0.0625 : 0.0576}

\text{i.e.}

{Q}_{B} \text{is maximum}

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