Earth as a System: Energy, Matter and Life - Describe solar radiation, electromagnetic spectrum, and differential heating of Earth

Calculate the frequency ($\nu$) of a green light wave part of the solar spectrum that has a wavelength ($\lambda$) of $500 \text{ nm}$. (Use $c = 3 \times 10^8 \text{ m/s}$ and $1 \text{ nm} = 10^{-9} \text{ m}$)

Given: $\lambda = 500 \times 10^{-9} \text{ m}$, $c = 3 \times 10^8 \text{ m/s}$. Using the formula: $\nu = \frac{c}{\lambda}$ $\nu = \frac{3 \times 10^8}{500 \times 10^{-9}}$ $\nu = 0.006 \times 10^{17} = 6 \times 10^{14} \text{ Hz}$

The frequency is inversely proportional to the wavelength. As the wavelength decreases, the frequency of the radiation increases.

Compare the heat absorbed by $1 \text{ kg}$ of dry land and $1 \text{ kg}$ of water to raise their temperature by $10^\circ C$. (Assume specific heat of land $c_{land} = 800 \text{ J/kg\cdot K}$ and water $c_{water} = 4184 \text{ J/kg\cdot K}$)

For Land: $Q_{land} = m \cdot c_{land} \cdot \Delta T = 1 \cdot 800 \cdot 10 = 8000 \text{ J}$ For Water: $Q_{water} = m \cdot c_{water} \cdot \Delta T = 1 \cdot 4184 \cdot 10 = 41840 \text{ J}$ Difference: $\begin{array}{r} 41840 \\ - 8000 \\ \hline 33840 \end{array}$

Water requires significantly more energy ($33,840 \text{ J}$ more) than land to reach the same temperature increase. This explains why oceans stay cooler than land during the day and warmer during the night.

Why is the solar intensity ($I$) lower at the poles than at the equator?

At the equator, the Sun is overhead, and the energy is concentrated over a small area: $I_{equator} = \frac{P}{A_{small}}$. At the poles, the same amount of solar energy is spread over a much larger area due to the angle of incidence: $I_{poles} = \frac{P}{A_{large}}$. Since $A_{large} > A_{small}$, then $I_{poles} < I_{equator}$.

The spherical shape of the Earth causes the 'Solar Footprint' to enlarge as you move toward the poles, resulting in less energy per square meter.