Chapter 3, Problem 1E

1. a. The Sun has an average surface temperature of 6000 K. How much radiation is emitted from this surface?
2. b. How much radiation is emitted from Earth’s surface at 300 K?

To determine

The amount of radiation emitted from the surface of the Sun, given the average surface temperature of Sun is 6000 K.

Answer to Problem 1E

The radiation emitted by the Sun at 6000K is $73,483,200{\text{Wm}}^{-\text{2}}$.

Explanation of Solution

Electromagnetic energy is radiated from the Sun, and it reaches the surface of the Earth in the form of visible radiations. Initially, the spectrum contains all types of wavelengths, and the atmosphere filters out certain wavelengths.

The emission of radiation by a body can be understood from Stefan–Boltzmann law. This law states, “Energy emitted by a body is directly proportional to the fourth power of temperature”.

$\begin{array}{l}E\propto T^4\\ E=\sigma T^4\end{array}$

Here, $E$ is the energy emitted by a body, T is the temperature of the body, and $\sigma$ is the Stefan–Boltzmann constant

The formula to calculate the radiation emitted by the Sun at 6,000K is expressed below:

$E=\sigma T^4$

Substitute $5.67\times {10}^{-8}{\text{Wm}}^{-\text{2}}{\text{K}}^{-\text{4}}$ for $\sigma$ and 6000 K for T to find the radiation emitted by the Sun at 6,000K.

$\begin{array}{c}E=\left(5.67\times {10}^{-8}{\text{Wm}}^{-\text{2}}{\text{K}}^{-\text{4}}\right){\left(6,000\text{K}\right)}^4\\ =73,483,200{\text{Wm}}^{-\text{2}}\end{array}$

Therefore, the radiation emitted by the Sun at 6000K is $73,483,200{\text{Wm}}^{-\text{2}}$.

To determine

The amount of radiation emitted from the Earth’s surface at 300 K.

Answer to Problem 1E

The radiation emitted by the Earth’s surface at 300K is $459.27{\text{Wm}}^{-\text{2}}$.

Explanation of Solution

Refer to subpart ‘a’ for description regarding Stefan–Boltzmann law.

The formula to calculate the radiation emitted by the Earth’s surface at 300K is expressed below:

$E=\sigma T^4$

To find the radiation emitted by the Earth’s surface at 300K, substitute $5.67\times {10}^{-8}{\text{Wm}}^{-\text{2}}{\text{K}}^{-\text{4}}$ for $\sigma$ and 300 K for T.

$\begin{array}{c}E=\left(5.67\times {10}^{-8}{\text{Wm}}^{-\text{2}}{\text{K}}^{-\text{4}}\right){\left(3,00\text{K}\right)}^4\\ =459.27{\text{Wm}}^{-\text{2}}\end{array}$

Therefore, the radiation emitted by the Earth’s surface at 300K is $459.27{\text{Wm}}^{-\text{2}}$.