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

Options

A. 225
B. 450
C. 900
D. 1800

Answer

D

Explanation

The energy radiated per second by a black body is given by Stefan's law,
`(E)/(t)sigmaT^(4)xxA`
where `A` is the surface area of the black body
`(E)/(t)=sigmaT^(4)xx4pir^(2)`
Since black body is a sphere, `A=4pir^(2)`
Case (i)
`(E)/(t)=450,T=500K,r=0.12m` ,
`:. 450=4pisigma(500)^(4)(0.12)^(2)` (i)
Case (ii)
`(E)/(t)=?,T=1000K,r=0.06m` (ii)
Dividing Eq. (ii) by Eq. (i), we get
`(E//t)/(450)=((1000)^(4)(0.06)^(2))/((500)^(4)(0.12)^(2))=(2^(4))/(2^(2))=4`
`implies (E)/(t)=450xx4=1800W`.