Solid angles and particle velocities

AI Thread Summary
To determine the fraction of particles in a gas traveling within a specific angular range, one must understand the concept of solid angles and their relation to the surface area of a sphere. The discussion highlights the need to calculate the area of a spherical cap, which represents a portion of the sphere's total surface area. The area can be derived through integration, although the exact formula was not recalled in the conversation. The shaded area in the provided diagram was noted to represent 20% of the sphere's total surface area. Understanding these concepts is crucial for accurately finding the angle theta related to the desired fraction of particle velocities.
quarky2001
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This should be a simple question, but I haven't found a clear explanation anywhere yet.

Suppose that there are a bunch of particles in a gas, with their velocities "uniformly distributed over solid angles", and I want to find out what fraction of particles are traveling with velocities in a certain angular range.

Conceptually, I know what this means. I'm just not sure how to do it - I drew a diagram.

Suppose the shaded area represents 20% of the sphere's total surface area. How do find the angle theta?

attachment.php?attachmentid=31859&stc=1&d=1296657266.jpg
 

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hi quarky2001! :smile:

(have a theta: θ :wink:)

that's called a cap of a sphere of radius θ …

i can't remember the formula for its area :redface:, but it's fairly easy to find by integration :wink:
 

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