Solid angles and particle velocities

Click For Summary
SUMMARY

The discussion focuses on calculating the fraction of particles in a gas that have velocities uniformly distributed over solid angles. The key concept involves determining the angular range corresponding to a specific fraction of the sphere's surface area, represented as a cap of a sphere. Participants mention the need for integration to derive the area formula for the spherical cap, which is essential for finding the angle theta (θ) that corresponds to the shaded area representing 20% of the sphere's total surface area.

PREREQUISITES
  • Understanding of solid angles and their distribution
  • Familiarity with spherical geometry
  • Basic knowledge of integration techniques
  • Concept of spherical caps and their properties
NEXT STEPS
  • Study the formula for the area of a spherical cap
  • Learn about solid angle calculations in three-dimensional space
  • Explore integration methods for geometric applications
  • Investigate particle velocity distributions in statistical mechanics
USEFUL FOR

Students and professionals in physics, particularly those studying gas dynamics, statistical mechanics, or geometric applications in three-dimensional space.

quarky2001
Messages
31
Reaction score
0
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
 

Attachments

  • Solidangle.jpg
    Solidangle.jpg
    13.8 KB · Views: 629
Physics news on Phys.org
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:
 

Similar threads

Finding area of a non right angled triangle
  • · Replies 9 ·
Replies
9
Views
4K
Undergrad How Is Particle Distribution in a Solid Angle Derived in Mechanics?
  • · Replies 3 ·
Replies
3
Views
1K
A ball is thrown w/initial speed vi at an angle 𝜃i with the horizontal
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Simple math model for a Particle Image Velocimetry system
  • · Replies 6 ·
Replies
6
Views
1K
Brownian Ratchet (exercise framed by Feynman's Lectures, l. 46)
  • · Replies 32 ·
2
Replies
32
Views
3K
A moving boat with a flag on its mast
  • · Replies 15 ·
Replies
15
Views
1K
Probability distribution of outgoing energy of particle
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Visualizing Solid Angle of a 3d Object (say a Sphere)
  • · Replies 6 ·
Replies
6
Views
3K
Determine the angular velocity as a function of the angle
  • · Replies 39 ·
2
Replies
39
Views
3K
Graduate Great! Glad to hear you were able to figure it out. Keep up the good work!
  • · Replies 7 ·
Replies
7
Views
3K