Complete Study Material for JEE, NEET, CBSE Board Class 11 Exams
In addition to the seven fundamental units in physics, two more units are considered supplementary units. These are Radian (rad) and Steradian (sr), which are defined as follows :
Plane Angle (dΞΈ)
It is defined as the ratio of the length of an arc of a circle to the radius of the circle.
Mathematically, dΞΈ = ds/r, where:
ds is the length of the arc
r is the radius of the circle
The SI unit of plane angle is the radian (rad).
Definition of Radian (rad)-First Supplementary Unit
1 radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
Relation between radian and degree : ΟΒ radians=180o
It is the three-dimensional analogue of plane angle and is defined as the area of a portion of the surface of a sphere divided by the square of the radius of the sphere.
Definition of Steradian(sr)-Second Supplementary Unit
1 steradian is the solid angle subtended at the center of a sphere by a surface of the sphere equal in area to a square having each side equal to the radius of the sphere.
A plane angle is the angle subtended at the center of a circle by an arc of the circle. It is measured in radians.
What is a solid angle?
A solid angle is the three-dimensional analogue of a plane angle and is subtended by a surface at the center of a sphere. It is measured in steradians.
How many steradians are there in a complete sphere?
A complete sphere subtends a solid angle of 4Ο steradians at its center.
Numerical Example
Example.1: What is the solid angle subtended by the moon at any point on Earth, given that the diameter of the moon is 3474 km and its distance from Earth is 3.84 Γ 10βΈ m?
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},
{
question: “What is the total solid angle subtended by a sphere?”,
options: [“2Ο sr”, “4Ο sr”, “6Ο sr”, “8Ο sr”],
answer: 1,
explanation: “A complete sphere subtends a total solid angle of 4Ο steradians at its center.”
}
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Do You Know?
The total solid angle around a point in space is 4Ο steradians.
1 radian β 57.3 degrees.
The concept of radian and steradian helps in understanding wave optics, astronomy, and electromagnetic field theory.
Worksheet
Define radian and steradian.
Convert 2 radians into degrees.
What is the solid angle subtended by the Sun at the Earth, given that the Sun’s diameter is 1.39 Γ 10βΉ m and its distance from the Earth is 1.496 Γ 10ΒΉΒΉ m?
Calculate the total solid angle subtended by a hemisphere.
Test Paper
Part A: Conceptual Questions
What is the difference between plane angle and solid angle?
How many radians are there in 90Β°?
Why is the unit steradian dimensionless?
Part B: Numerical Questions
The radius of a circular disc is 10 cm. Find the angle subtended by a 5 cm arc at the center.
The surface area of a sphere is 1256 cmΒ². What is the total solid angle subtended at its center?
Important Points for Quick Revision
Plane angle is measured in radians (rad).
Solid angle is measured in steradians (sr).
1 radian = 180Β°/Ο β 57.3Β°.
1 steradian is the angle subtended by an area equal to rΒ² on a sphere.
The total solid angle around a point is 4Ο sr.
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