Energy from the Sun received at the Earth cross section

[Puchinita5]

Homework Statement

My homework involves calculating the amount of energy the Earth receives from the Sun in a given year. I actually have the answer, but what I'm confused with is that the calculation says that the Sun "sees the cross section of the Earth, a circle". So in the calculation, we multiply the energy per square meter received at 1AU by pi*R^2 (R is the radius of the Earth).

Why isn't it multiplied by half of the surface area of Earth? I picture half of the Earth facing the sun as the surface receiving the energy. So I would think to multiply by 2*pi*r^2.

Homework Equations

The Attempt at a Solution

 

[BvU]

Hi.

$\pi r^2$ is the area of a circle. That's what the sun 'sees'.

You seem to be thinking of half a sphere, but then

1. a sphere has area ${4\over 3} \pi r^2$
2. the angle of incidence has to be corrected for -- which brings you back to $\pi r^2$

Make a little sketch showing this angle of incidence

Or check the shadow of a sphere on a wall !

 

[haruspex]

Why isn't it multiplied by half of the surface area of Earth?

Because the Sun doesn't care what shape the Earth is. It sends out the same power per unit solid angle in all directions. The power the Earth gets is proportional to the solid angle it subtends at the Sun.