Chapter 1.11, Problem 43E

Radiation Energy The total radiation energy E emitted by a heated surface per unit area varies as the fourth power of its absolute temperature T. The temperature is 6000 K at the surface of the sun and 300 K at the surface of the earth.

1. (a) How many times more radiation energy per unit area is produced by the sun than by the earth?
2. (b) The radius of the earth is 3960 mi, and the radius of the sun is 435,000 mi. How many times more total radiation does the sun emit than the earth?

To determine

To find: The number of times more radiation energy per unit area emitted by the sun than the earth.

Answer to Problem 43E

The radiation energy per unit area emitted by sun is greater than earth by factor of $160,000$.

Explanation of Solution

Given:

The total energy per unit area emitted by a heated surface is directly proportional to fourth power of its absolute temperature.

Temperature of surface of the sun is $6000\text{K}$.

Temperature of the surface of the earth is $300\text{K}$.

Calculation:

The total radiation energy $\left(E\right)$ emitted per unit surface area by the heated object is directly proportional to the fourth power of its absolute temperature.

That is,

$\frac{E}{A}=k{T}^4$ (1)

Here,

$E$ is the total radiation energy emitted by the heated surface.

$A$ is the surface area.

$T$ is the absolute temperature of the heated surface.

$k$ is the constant of proportionality.

To calculate Radiation energy $\left(E_1\right)$ emitted per unit surface area by the sun, substitute $1$ for $A$ and $6000$ for $T$ in equation (1).

$\begin{array}{l}\frac{E_1}{\left(1\right)}=k{\left(6000\right)}^4\\ E_1=k{\left(6000\right)}^4\end{array}$ (2)

To calculate Radiation energy $\left(E_2\right)$ emitted per unit surface area by the earth, substitute $1$ for $A$ and $300$ for $T$ in equation (1).

$\begin{array}{l}\frac{E_2}{\left(1\right)}=k{\left(300\right)}^4\\ E_2=k{\left(300\right)}^4\end{array}$ (3)

Divide equation (2) by equation (3).

$\begin{array}{l}\frac{E_1}{E_2}=\frac{k{\left(6000\right)}^4}{k{\left(300\right)}^4}\\ \frac{E_1}{E_2}={\left(\frac{6000}{300}\right)}^4\\ \frac{E_1}{E_2}={\left(20\right)}^4\\ \frac{E_1}{E_2}=160000\end{array}$

From the above equation,

$E_1=160000E_2$

Thus, the radiation energy emitted per unit surface area is greater than earth by factor of $160,000$.

To determine

To find: The number of times more total radiation emitted by the sun than the earth.

Answer to Problem 43E

The total radiation energy emitted by the sun is greater than the earth by the factor of $1,930,670,340$.

Explanation of Solution

Given:

Radius $\left(r_1\right)$ of the earth is $3960\text{mi}$.

Radius $\left(r_2\right)$ of the sun is $435,000\text{mi}$.

Temperature of surface of the sun is $6000\text{K}$.

Temperature of the surface of the earth is $300\text{K}$.

Calculation:

Consider both earth and the sun as sphere.

Surface area of the sphere is $4\pi r^2$.

Here, $r$ is the radius of the sphere.

Use equation (1) and calculate total radiation energy emitted by the sun.

Substitute $4\pi r_1^2$ for $A$, $6000$ for $T$ in equation (1) and calculate total energy emitted by the sun.

$\frac{E_{\text{Sun}}}{4\pi {\left(r_1\right)}^2}=k{\left(6000\right)}^4$ (4)

Now, use equation (1) and calculate total radiation energy emitted by the earth

Substitute $4\pi r_2^2$ for $A$, $300$ for $T$ in equation (1) and calculate total energy emitted by the earth.

$\frac{E_{\text{earth}}}{4\pi {\left(r_2\right)}^2}=k{\left(300\right)}^4$ (5)

Divide equation (4) by equation (5) and simplify.

$\begin{array}{l}\frac{\frac{E_{\text{Sun}}}{4\pi {\left(r_1\right)}^2}}{\frac{E_{\text{earth}}}{4\pi {\left(r_2\right)}^2}}=\frac{k{\left(6000\right)}^4}{k{\left(300\right)}^4}\\ \frac{E_{\text{Sun}}}{E_{\text{earth}}}\cdot \left(\frac{r_2^2}{r_1^2}\right)={\left(\frac{6000}{300}\right)}^4\end{array}$

Now, substitute $3960$ for $r_2$ and $435000$ for $r_1$ in above equation.

$\begin{array}{l}\frac{E_{\text{Sun}}}{E_{\text{earth}}}\cdot \left(\frac{{3960}^2}{{435000}^2}\right)={\left(\frac{6000}{300}\right)}^4\\ \frac{E_{\text{Sun}}}{E_{\text{earth}}}\cdot \left(8.28\times {10}^{-5}\right)=160000\\ \frac{E_{\text{Sun}}}{E_{\text{earth}}}=\frac{160000}{8.28\times {10}^{-5}}\\ \frac{E_{\text{Sun}}}{E_{\text{earth}}}=1932367150\end{array}$

Thus, the total energy emitted by the sun is greater than the earth by the factor of approximately $1,930,670,340$.