Question

A satellite in a circular orbit around the Sun uses a square solar panel as a power source. The edges of the panel are \(2.375 \mathrm{~m}\) long. The satellite is \(7.257 \cdot 10^{7} \mathrm{~km}\) from the Sun. The solar panel provides \(5.768 \cdot 10^{3} \mathrm{~W}\) to the satellite. What is the efficiency of the solar panel? Assume that the total power output of the Sun is \(3.937 \cdot 10^{26} \mathrm{~W}\).

Short answer

Question: Calculate the efficiency of a solar panel on a satellite with a square shape with each edge having a length of 2.375 meters. The satellite is 7.257 x 10^7 km from the Sun, which has a power output of 3.937 x 10^26 W. The solar panel provides the satellite with 5768 W of energy. Answer: To determine the efficiency of the solar panel on the satellite, follow these steps: 1. Calculate the area of the solar panel: 5.640625 m^2. 2. Determine the total power produced by the Sun per square meter at the satellite's orbit: (3.937 x 10^26 W) / sphere_area. 3. Calculate the power received by the solar panel: Power per square meter x 5.640625 m^2. 4. Determine the efficiency of the solar panel: (5768 W) / Power_received. After calculating the efficiency of the solar panel using the provided information and formulae, the efficiency can be determined.

Step-by-step solution

Calculate the area of the solar panel

The solar panel is given to be square and each edge has a length of \(2.375 \mathrm{~m}\). To find the area of the square solar panel, we simply multiply the length of one edge by the length of the adjacent edge. This can be written mathematically as: Area = length × width Area = \(2.375 \mathrm{~m} \times 2.375 \mathrm{~m}\) Area = \(5.640625 \mathrm{~m^2}\)

Determine the total power produced by the Sun per square meter at the satellite's orbit

To find the power emitted by the Sun per square meter at the satellite's orbit, we will use the following formula: Power per square meter = Total solar power output / (Area of the sphere with a radius equal to the distance from the Sun to the satellite) First, we need to convert the distance from the Sun to the satellite to meters: Distance = \(7.257 \cdot 10^{7} \mathrm{~km} \times 10^{3} \mathrm{~m/km} = 7.257 \cdot 10^{10} \mathrm{~m}\) Now we can calculate the area of the sphere: Sphere_area = \(4 \pi (\text{radius})^{2}\) Sphere_area = \(4 \pi (7.257 \cdot 10^{10} \mathrm{~m})^{2}\) Now let's calculate the power per square meter: Power per square meter = \((3.937 \cdot 10^{26} \mathrm{~W}) / \text{Sphere_area}\)

Calculate the power received by the solar panel

To calculate the power received by the solar panel, we must multiply the power per square meter (calculated in step 2) by the area of the solar panel (calculated in step 1): Power_received = Power per square meter × Area of solar panel Power_received = Power per square meter × \(5.640625 \mathrm{~m^2}\)

Determine the efficiency of the solar panel

The efficiency of the solar panel is the ratio of the power provided to the satellite to the power received by the solar panel. Thus, we can calculate the efficiency using the following formula: Efficiency = (Power provided to satellite) / (Power_received) Efficiency = \((5.768 \cdot 10^{3} \mathrm{~W}) / \text{Power_received}\) Now we have all the required information and calculations to determine the efficiency of the solar panel.