Question

The solar power striking Earth every day averages 168 watts per square meter. The peak electrical power usage in New York City is 12,000 megawatts. Considering that present technology for solar energy conversion is only about \(10 \%\) efficient, from how many square meters of land must sunlight be collected in order to provide this peak power? (For comparison, the total area of the city is \(830 \mathrm{~km}^{2}\).)

Short answer

Approximately \(7.14 \times 10^8\) square meters of land must be used to generate the peak power for New York City using the given solar energy conversion efficiency of 10%.

Step-by-step solution

Find power generated by 1 square meter with given efficiency.

We are given that the average solar power striking Earth every day is 168 watts per square meter. The solar energy conversion efficiency is 10%. We need to find the power generated by 1 square meter with this efficiency. Power generated per square meter = (Solar Power per square meter) × (Conversion efficiency) Power generated per square meter = \(168 \mathrm{W/square~meter} \times 0.1\) Power generated per square meter = 16.8 watts

Convert New York City's peak power usage to watts.

We are given that New York City's peak power usage is 12,000 megawatts. We can convert this to watts for easier calculations. 1 megawatt (MW) = 1,000,000 watts = \(10^6 \mathrm{W}\) 12,000 megawatts = 12,000 × \(10^6 \mathrm{W} = 1.2 \times 10^{10} \mathrm{W}\) The peak power usage in New York City is \(1.2 \times 10^{10} \mathrm{W}\).

Divide the desired power by the power generated per square meter.

Now, we will find the number of square meters required to generate the desired power by dividing the desired power (peak power usage) by the power generated per square meter. Number of square meters = \(\frac{1.2 \times 10^{10} \mathrm{W}}{16.8 \mathrm{W/square~meter}}\) Number of square meters = \(\frac{1.2 \times 10^9}{16.8}\) Number of square meters = \(7.14 \times 10^{8}\) Therefore, approximately \(7.14 \times 10^8\) square meters of land must be used to generate the peak power for New York City using the given solar energy conversion efficiency.

Key concepts

Solar Power

Understanding solar power is crucial for the potential harnessing of the sun's energy. Solar power refers to the energy from the sun that is converted into thermal or electrical energy. The sun emits energy in the form of photons, which solar panels capture and convert into usable power.

Solar panels consist of many solar cells made of semiconductor materials, such as silicon. When sunlight hits a solar cell, the energy from the photons knocks electrons loose from the atoms in the semiconductor material. An electric circuit is formed, and these free electrons flow through it, generating electricity.

It's important to note that the amount of solar power striking Earth varies depending on the time of day, weather conditions, and geographic location. The given exercise uses the average daily solar power measured in watts per square meter, a standard way of quantifying solar irradiance. In our case, knowing that the Earth receives 168 watts per square meter on average helps us in our calculations for the required land to generate sufficient electricity.

Energy Conversion

Energy conversion efficiency is a measure of a system's ability to convert one form of energy into another, particularly useful form. For solar panels, it's the rate at which they convert sunlight into electricity. The efficiency of current solar technology plays a significant role in determining how much solar energy we need to harness to meet power demands.

As solar technology advances, the efficiency of solar panels is also improving. However, for this exercise, we are considering a conversion efficiency of 10%. This percentage tells us what fraction of the incoming solar energy is actually being turned into electrical power the panels produce. With this in mind, when we multiply the average solar power per square meter by the efficiency rate, we get the actual power output per square meter of solar panel.

Understanding that efficiency rates vary and improving them could reduce the need for larger solar farms is key to grasping the broader potential of solar power. Increasing efficiency helps to minimize the land and resources required to produce a given amount of electricity, which is especially significant in dense urban areas like New York City.

Power Calculation

Power calculation is essential to understand how we can meet electricity demands using solar energy. It involves determining the amount of electrical power generated or needed and is expressed in watts (W). The calculation in our exercise is to find out how many square meters of solar panels would be required to meet New York City's peak power usage.

The process begins by figuring out the power output of a single square meter of solar panel, given its efficiency. Multiplying the solar power striking Earth per square meter by the energy conversion efficiency provides us with this value. Next, we take the total power needed – in this case, New York City's peak power usage – and convert it into watts to align with our other measurements.

Finally, by dividing the total power requirement by the power output per square meter, we determine the total area of solar panels needed. This type of calculation helps us to plan for the infrastructure required to support solar energy as a substantial source of power, especially in large urban areas. Moreover, improvements in solar panel efficiency would directly decrease the amount of land needed for solar farms, making these calculations even more pivotal.