A spherical black body with a radius of 12 cm radiates 450 W power at 50 K. If t

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Question

A spherical black body with a radius of 12 cm radiates 450 W power at 50 K. If the radius was halved and the temperature doubled, the power radiated in watt would be:

Options:

A . 225

B . 450

C . 900

D . 1800

Answer: Option D
:
D
The energy radiated per second by a black body is given by Stefan's Law as

\frac{E}{t} = σ T^{2} \times A

where A is the surface area of the black body

\frac{E}{t} σ T^{4} \times 4 π r^{2}

Since black body is a sphere,

(A = 4 π r^{2})

.
Case(i)

\frac{E}{t} = 450, T = 500 K, r = 0.12 m,

∴ 450 = 4 π σ (500)^{4} (0.12)^{2}

.....(i)
Case(ii)

\frac{E}{t} = ?, T = 1000 K, r = 0.06 m,

dividing (ii) and (I) we get,

\frac{\frac{E}{t}}{450} = \frac{(1000)^{4} (0.06)^{2}}{(500)^{4} (0.12)^{2}} = \frac{2^{4}}{2^{2}} = 4

\Rightarrow \frac{E}{t} = 450 \times 4 = 1800 W

(d) is the correct option.

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