Understanding Solid Angle and its Relation to s=rθ

In summary, 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. It is similar to the definition of angle, theta=s/R, where s is an arc length and R is the radius of a circle. Both angle and solid angle are dimensionless, with radians and steradians used for convenience. The use of R^2 in the definition of solid angle makes it independent of the size of the sphere, just like how dividing s by R for radians makes it independent of the size of the circle. This is a simple mathematical concept, not specific to any physics.
  • #1
hasanhabibul
31
0
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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  • #2
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.
 
  • #3
why u r using r^2..it is just to make the whole thing dimestionless or any other physics in it..??
 
  • #4
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.
 

1. What is solid angle and why is it important for understanding s=rθ?

Solid angle is a fundamental concept in mathematics and physics that measures the amount of space covered by a three-dimensional object as seen from a given point. It is important for understanding s=rθ because it helps us calculate the surface area (s) of a sphere, given its radius (r) and central angle (θ).

2. How is solid angle related to the radius and central angle of a sphere?

The solid angle (Ω) is directly proportional to both the radius (r) and the central angle (θ) of a sphere. This means that as either the radius or the central angle increases, the solid angle also increases.

3. Can the solid angle of a sphere be greater than 4π steradians?

No, the solid angle of a sphere cannot be greater than 4π steradians. This is because 4π steradians is the maximum possible solid angle that can be covered by a sphere, which is equivalent to the surface area of the entire sphere.

4. How is solid angle different from regular angle?

Solid angle is a measure of the amount of space covered by a three-dimensional object, while regular angle measures the amount of rotation between two intersecting lines. Additionally, solid angle is a three-dimensional concept, while regular angle is a two-dimensional concept.

5. What are some real-life applications of understanding solid angle and its relation to s=rθ?

Understanding solid angle and its relation to s=rθ has various real-life applications, such as in computer graphics, optics, and astronomy. It is used to calculate the field of view of cameras and lenses, to design and optimize lighting systems, and to analyze the radiation patterns of antennas and satellites.

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