Understanding Solid Angle and its Relation to s=rθ

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Discussion Overview

The discussion revolves around the concept of solid angle and its mathematical relationship to the formula s = rθ, exploring the definitions and dimensionality of angles and solid angles in the context of geometry and physics.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant questions why solid angle is defined as A/r² and seeks to understand the reasoning behind the r² term.
  • Another participant explains that solid angle is analogous to the angle defined by arc length and radius, stating that both angles are dimensionless and introducing the terms radian and steradian.
  • A participant challenges the necessity of the r² term, asking if it serves merely to make solid angle dimensionless or if there are other physical implications.
  • It is suggested that the division of arc length by radius for radians is due to the proportionality of arc length to radius, and a similar reasoning applies to solid angles, where area is proportional to r².

Areas of Agreement / Disagreement

Participants express differing views on the reasons for the r² term in solid angle, with some agreeing on the mathematical relationships while others question the underlying physics or implications.

Contextual Notes

Participants note that the definitions of angles and solid angles may depend on the context, particularly when considering surfaces that are not spherical, which introduces complexity in the definition of solid angle.

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why solid angle is A/r^2 ...why is this r^2...has it any similarity with s=rtheta??please help me
 
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They are similar.
Angle is defined as theta=s/R, where s is an arc length and R the radius of a circle.
Solid angle is defined as Omega=A/R^2, where A is an area on a sphere and R is the radius of the sphere.
In each case, the angle and solid angle are dimensionless, but given the names radian and steradian for convenience.
If the surface is not on a sphere, then differential vectors must be used in the definition of solid angle.
 
why u r using r^2..it is just to make the whole thing dimestionless or any other physics in it..??
 
It does make solid angle dimensionless, but there are other reasons too.
Why divide s by R for radians? Because the arc length is proportional to R.
Dividing the arc length by R makes the angle measure in radians independent of the size of the circle. The same reasoning gives R^2 for steradians, because the area isl proportional to R^2. This is all simple mathematics, independent of any physics.
 

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