Calculating steradians (solid angle)

[jmc]

Homework Statement

For a sphere of radius r, find the solid angle Ω in steradians defined by spherical angles
of: a.) 0°≤θ≤ 20°, 0°≤ø≤360°;

Homework Equations

dA = r² sin dθ dø (m²)
dΩ = dA / r² = sin dθ dø (sr)

The Attempt at a Solution

I think I understand what a steradian (sr) is, on a sphere, and I need to find what ratio of a 4∏ is formed by the above limits, but I can't make the connection on how to form that ratio. ?

 

[SteamKing]

First, draw a sketch of the region. I assume you are familiar with spherical coordinates?

 

[jmc]

I have tried to sketch it. What I see is.the top portion of the sphere, sliced where 20 deg from the X-axis meets the edge of the sphere.
But how to convert that to a steridian, I still don't see.

Thank you

 

[SteamKing]

I have tried to sketch it. What I see is.the top portion of the sphere, sliced where 20 deg from the X-axis meets the edge of the sphere.
But how to convert that to a steridian, I still don't see.

Thank you

You have to go back to the definition of what the steradian represents:

http://en.wikipedia.org/wiki/Steradian

In other words, you got some calculatin' to do. You know the region for which the steradian is desired, now you have to calculate the values of the quantities in the formula for the steradian.

 

[nrqed]

Homework Statement

For a sphere of radius r, find the solid angle Ω in steradians defined by spherical angles
of: a.) 0°≤θ≤ 20°, 0°≤ø≤360°;

Homework Equations

dA = r² sin dθ dø (m²)
dΩ = dA / r² = sin dθ dø (sr)

The Attempt at a Solution

I think I understand what a steradian (sr) is, on a sphere, and I need to find what ratio of a 4∏ is formed by the above limits, but I can't make the connection on how to form that ratio. ?

You do not need to take any ratio. You simply have to integrate $d \Omega$ over the section you are considering.