A spherical black body with a radius of 12 cm radiates 450 watt power at 500 K. If the radius were halved and the temperature doubled, the power radiated in watt would be [2017]

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Solution

Q.
A spherical black body with a radius of 12 cm radiates 450 watt power at 500 K. If the radius were halved and the temperature doubled, the power radiated in watt would be [2017]

1 450

2 1000

3 1800

4 225

Ans.
(3)

According to Stefan-Boltzmann law, rate of energy radiated by a black body is given as

$E = σ A T^{4} = σ 4 π R^{2} T^{4}$

Given $E_{1} = 450 W, T_{1} = 500 K, R_{1} = 12 cm$

$R_{2} = \frac{R_{1}}{2}, T_{2} = 2 T_{1}, E_{2} = ?$

$\frac{E_{2}}{E_{1}} = \frac{σ 4 π R_{2}^{2} T_{2}^{4}}{σ 4 π R_{1}^{2} T_{1}^{4}} = {(\frac{R_{2}}{R_{1}})}^{2} {(\frac{T_{2}}{T_{1}})}^{4} \Rightarrow \frac{E_{2}}{E_{1}} = \frac{1}{4} \times 16 = 4$

$E_{2} = E_{1} \times 4 = 450 \times 4 = 1800$ $W$

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