Question

The intensity of electromagnetic waves from the Sun at the Earth is \(1400 \mathrm{~W} / \mathrm{m}^{2}\). How much power does the Sun generate? The distance from the Earth to the Sun is \(149.6 \cdot 10^{6} \mathrm{~km}\). a) \(1.21 \cdot 10^{20} \mathrm{~W}\) b) \(2.43 \cdot 10^{24} \mathrm{~W}\) c) \(3.94 \cdot 10^{26} \mathrm{~W}\) d) \(2.11 \cdot 10^{28} \mathrm{~W}\) e) \(9.11 \cdot 10^{30} \mathrm{~W}\)

Short answer

Answer: (c) \(3.94 \cdot 10^{26} \mathrm{~W}\)

Step-by-step solution

Determine the known values

We are given the intensity of the Sun's electromagnetic waves at Earth (\(I = 1400 \mathrm{~W/m^2}\)) and the distance between the Earth and the Sun (\(d = 149.6 \cdot 10^6 \mathrm{~km}\)).

Convert the distance to meters

Before proceeding, we need to convert the distance from kilometers to meters. \(d = 149.6 \cdot 10^6 \mathrm{~km} \cdot \frac{1000 \mathrm{~m}}{1 \mathrm{~km}} = 149.6 \cdot 10^9 \mathrm{~m}\)

Use the formula for intensity

The intensity of a point source at a given distance is given by the formula \(I = \frac{P}{A}\), where \(I\) is the intensity, \(P\) is the power of the source, and \(A\) is the surface area of a sphere centered at the source point. Rearrange the formula to solve for the power: \(P = I \cdot A\)

Calculate the surface area of the sphere

The surface area of a sphere is given by the formula \(A = 4 \pi r^2\), where \(r\) is the radius (in this case, the distance from the Sun to the Earth). So the surface area is \(A = 4 \pi (149.6 \cdot 10^9)^2\).

Plug in the values and solve for the power

Using the values from steps 1 and 4, we can now solve for the power generated by the Sun: \(P = I \cdot A = 1400 \mathrm{~W/m^2} \cdot 4 \pi (149.6 \cdot 10^9)^2 \approx 3.94 \cdot 10^{26} \mathrm{~W}\) So, the correct answer is (c) \(3.94 \cdot 10^{26} \mathrm{~W}\).

Key concepts

Solar Power Calculation

Understanding solar power calculation is essential when you're analyzing the enormous energy that the Sun provides. The process begins with a critical parameter: the intensity of electromagnetic waves given at a certain distance from the Sun.

In our case, we're considering the intensity of sunlight when it reaches the Earth, which is on average about 1400 Watts per square meter (\(1400 \text{ W/m}^2\)). Remember this figure represents the power distributed over an area of one square meter. With this information, you might wonder how this translates to the total power output of the Sun.

To convert the intensity into total power, you need to think of the Sun's energy spreading out uniformly in all directions, forming a spherical shell at the distance of the Earth's orbit. By calculating the total area of this shell (using the surface area formula of a sphere) and then multiplying it by the intensity, you can find out the total power of the Sun.

It's worth noting that in an actual scenario, one must account for factors such as the Earth's atmosphere and its albedo, which reflect and absorb some of the Sun's energy. However, for the purpose of this exercise, we assume a vacuum between the Sun and the Earth to simplify our calculations.

Intensity of Electromagnetic Waves

Intensity is a measure of the power per unit area carried by a wave. In the context of sunlight, it refers to the amount of solar energy that hits a given area per second. Electromagnetic waves, such as sunlight, spread out as they travel through space, and their intensity decreases with the square of the distance from their source due to the conservation of energy over the surface area of an expanding sphere.

The intensity formula is given by \(I = \frac{P}{A}\), where \(I\) is the intensity, \(P\) is the power output of the source, and \(A\) is the surface area of the sphere through which the power is spread. This relationship enables us to solve for any of the variables if the other two are known. It also illustrates why the intensity of sunlight is much greater on Earth's surface than further out in space, as the same energy is spread over a larger area.

Surface Area of a Sphere

The surface area of a sphere is a key concept when working with any sphere-related problem, such as calculating the power generated by the Sun. The formula for the surface area of a sphere is \(A = 4\pi r^2\), where \(r\) represents the radius of the sphere.

When the Sun emits energy, it does so in all directions equally, forming an imaginary sphere at a certain distance. The surface area of this sphere increases quadratically with the radius, which in our case is the distance from the Sun to the Earth.

Therefore, to find out this vast area through which the Sun's power is spread, you square the distance from the Sun to the Earth and then multiply by \(4\pi\). This calculation produces an impressively large number, reflecting the expansive reach of the Sun's energy. By understanding the surface area of a sphere, we've bridged the gap between a local intensity measurement and the total power output of a star like the Sun.