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Chapter 1: Problem 87

The total rate at which power is used by humans worldwide is approximately 15 TW (terawatts). The solar flux averaged over the sunlit half of Earth is $680 \mathrm{~W} / \mathrm{m}^{2}$ (assuming no clouds). The area of Earth's disc as seen from the Sun is \(1.28 \times 10^{14} \mathrm{~m}^{2}\). The surface area of Earth is approximately 197,000,000 square miles. How much of Earth's surface would we need to cover with solar energy collectors to power the planet for use by all humans? Assume that the solar energy collectors can convert only \(10 \%\) of the available sunlight into useful power.

Short Answer

Expert verified
Approximately 4.32% of Earth's surface would need to be covered with solar energy collectors to power the entire planet for use by all humans.

Step by step solution

01

Calculate Total Solar Power Received by Earth

First, we need to find the total amount of solar power received by the Earth. We have the average solar flux over the sunlit half of Earth (680 W/m²) and the area of Earth's disc as seen from the Sun (1.28 x 10^14 m²). Total solar power received by Earth = Average solar flux x Area of Earth's disc = 680 W/m² x 1.28 x 10^14 m² = 8.704 x 10^16 W
02

Calculate Power Received by Solar Collectors

Now, we need to calculate the amount of power that can be collected by solar energy collectors. They can convert only 10% of the available sunlight into useful power. Therefore, the power received by the solar energy collectors is 10% of the total power received by Earth. Power received by solar collectors = 10% of 8.704 x 10^16 W = 0.10 x 8.704 x 10^16 W = 8.704 x 10^15 W
03

Calculate the Required Power Generated by Solar Collectors

We are given that the total rate at which power is used by humans worldwide is 15 TW. Thus, we need the solar energy collectors to generate this amount of power. 1 TW = 10^12 W, so 15 TW = 15 x 10^12 W
04

Calculate the Required Area of Solar Collectors

Now let's determine the area required to generate the power needed by all humans. We have the power received by solar collectors and the required power. Area required = (Power required by humans) / (Power received by solar collectors per square meter) Area required = (15 x 10^12 W) / (680 W/m²) = 2.20588 x 10^10 m²
05

Convert Square Meters to Square Miles

We need to convert the area in square meters to square miles. 1 square meter is equal to approximately 3.86102 x 10^-7 square miles. Area required in square miles = 2.20588 x 10^10 m² x 3.86102 x 10^-7 mi/m² ≈ 8,521,621 square miles
06

Calculate the Percentage of Earth's Surface

Finally, let's compute the percentage of Earth's surface that needs to be covered by solar panels. Percentage = (Required area of solar panels) / (Total surface area of Earth) x 100 Percentage = (8,521,621 square miles) / (197,000,000 square miles) x 100 ≈ 4.32 % Approximately 4.32% of Earth's surface would need to be covered with solar energy collectors to power the entire planet for use by all humans.

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