Question

The surface of the earth receives solar radiation at the rate of $1400W{m}^{-2}$. The distance of the centre of the sun from the surface of the earth is $1.5\times {10}^{11}m$ and the radius of the sun is $7.0\times {10}^{8}m$. Treating the sun as a black body, it follows from the above data that the surface temperature of the sun is about

• A. $58000K$
• B. $5800K$
• C. $6000K$
• D. $6200K$

Answer 1

Answer: (B)

$5800K$

The energy radiated from a blackbody from Stefan's law is $E=\sigma A{T}^4$
For sun $E=4\pi R^2\sigma T^4$
where $\sigma =5.67\ast {10}^{-8}$ is Stefan's constant, T is the surface temp of the blackbody in Kelvin and A is the surface area of the blackbody.
The solid angle subtended by a unit surface area on earth's surface at sun's center is $d\Omega =\frac{1}{4\pi r^2}$ where r is the distance from sun's center to earth.
Given,
$E\ast d\Omega =1400$
$R=7\ast {10}^8$
$r=1.5\ast {10}^{11}$
$4\pi R^2\sigma T^4\ast \frac{1}{4\pi r^2}=\frac{R^2}{r^2}\ast \sigma T^4=1400$
$\therefore T^4\ast \frac{49}{2.25}\ast {10}^{-6}\ast 5.67\ast {10}^{-8}=1400$
$\therefore T^4=.1134\ast {10}^{16}$
$\therefore T=5.83\ast {10}^3=5830$