Question

Imagine the Earth is on the surface of a sphere whose radius is equal to the sun-Earth distance, \(1.50 \times 1011 \mathrm{~m}\). If the total radiant flux from the sun (W) calculated above is evenly distributed over the surface of that imaginary sphere, what is the average flux in \(\mathrm{W} \mathrm{m}^{-2}\) ?

Short answer

Answer: The equation to calculate the average flux on Earth's surface in terms of the total radiant power (P) from the sun is \(F_{avg} = \frac{P}{9\times 10^{22}\pi}\) W m\(^{-2}\).

Step-by-step solution

Determine the total radiant power from the sun

We are not given the total radiant power from the sun in the problem. However, we know that this value is necessary for our calculation. Let's denote this value as P (in Watts).

Calculate the imaginary sphere's surface area

We are given the radius of the sphere \(r = 1.5 \times 10^{11}\) m. To calculate the surface area of this imaginary sphere, we will use the formula: \(A = 4 \pi r^2\) Substituting the given value of the radius, we get: \(A = 4 \pi (1.5 \times 10^{11})^2\) \(A = 4 \pi (2.25 \times 10^{22})\) \(A = 9 \times 10^{22}\pi\) m\(^2\)

Calculate the average flux

Now, we need to calculate the average flux on the Earth's surface, denoted as \(F_{avg}\), using the total radiant power (P) and the surface area (A) that we gained in the previous steps: \(F_{avg} = \frac{P}{A}\) As we don't have the value of P from the problem text, let's rewrite this equation using P: \(F_{avg} = \frac{P}{9\times 10^{22}\pi}\) W m\(^{-2}\) The equation above represents the average flux on the Earth's surface in terms of total radiant power (P).