Chapter 22, Problem 65RPP

Rainbows How is a rainbow formed? Recall that the index of refraction of a medium is slightly different for different colors. When white light from the Sun enters a spherical raindrop, as shown in Figure 22.25, the light is refracted, or bent After reflecting off the back surface of the drop, the light is refracted again as it leaves the front surface.

Each drop separates the colors of light. An observer on the ground with her back to the Sun sees at most one color of light coming from a particular drop (see Figure 22.26). If the observer sees rod light from a drop (for example, the top drop in Figure 22.26). the violet light for that same drop is deflected above her head. However, if she sees violet light coming from a drop lower in the sky the red light from that drop is defected below her eyes onto the ground She sees red light when her line of view makes an angle of $42°$ with the beam of sunlight and violet light when the angle is $40°$, Other colors of light are seen at intermediate angles.

Earth energy balance Gases in Earth's atmosphere, such as carbon dioxide and water vapor act like a blanket that reduces the amount of energy that Earth radiates into space. This phenomenon is called the greenhouse effect. Without the greenhouse effect most of Earth would have a climate comparable to that of the polar or subpolar regions What would Earth's mean surface temperature be, in the absence of the gases causing the greenhouse effect?

The Sun continually irradiates our upper atmosphere with an intensity of about ${\text{1360 W/m}}^{\text{2}}\text{.}$

About $\text{30\%}$ of this energy is reflected back into space, leaving an average of ${\text{950 W/m}}^{\text{2}}$ to be absorbed by the surface area of Earth that is exposed to the Sun This exposed area is $\pi R^2{}_{\text{Earth}},$ the Earth s circular cross section that intercepts the sunlight All objects, including Earth, emit radiation at a rate proportional to the fourth power of the surface temperature $\left(T^4\right)$ in kelvins $\left(K\right)$. Thus, the radiation is emitted from Earth's surface, which as a sphere has the area $4\pi R^2{}_{\text{Earth}}$.

To maintain a constant temperature, Earth's radiation rate must equal its energy absorption rate from the Sun A fairly simple calculation indicates that the two rates are equal when the average surface temperature of Earth is 255 K or about $\text{0 °F}$. However, Earth s mean surface temperature is much warmer than that—about 288 K—because the calculation neglected the effect of greenhouse gases in the atmosphere. These gases absorb infrared radiation emitted by Earth and reflect some of it back to Earth Thus Earth emits less energy into space than it would without greenhouse gases. The energy absorbed from the Sun and the reduced energy emitted into space have caused the Earth to warm to its mean surface temperature of 288 K.

Over the past two centuries the concentration of carbon dioxide in our atmosphere has increased from a pre- industrial level of about 270 parts per million to 380 parts per million. This increase in carbon dioxide and other greenhouse gases has been caused by the burning of fossil fuels and the removal of forests, which absorb carbon dioxide. The carbon dioxide concentration in the atmosphere is expected to reach 600-700 parts per million by 2100. If that occurs, it will be warmer in 2100 than at any time in the last half million years.

The Sun irradiates Earth’s outer atmosphere at what rate?

$\begin{array}{l}\text{a}\text{. 1}\times {\text{ 10}}^{\text{16}}\text{J/s}\hfill \\ \text{b}\text{. 4 }\times {\text{ 10}}^{\text{16}}\text{J/s}\hfill \\ \text{c}\text{. 2 }\times {\text{ 10}}^{\text{17}}\text{J/s}\hfill \\ \text{d}\text{. 7 }\times {\text{ 10}}^{\text{17}}\text{J/s}\hfill \\ \text{e}\text{. 1400 J/s}\hfill \end{array}$