Supplementary Units (Radian & Steradian) Class 11 Physics | Question-Answer, FAQS, Worksheet, MCQS, Test Paper

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 (dθ)

It is defined as the ratio of the length of an arc of a circle to the radius of the circle.

Mathematically, dθ = 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 : π radians=180^o

Solid Angle (dΩ)

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.

Mathematically, dΩ = dA/r², where:

dA is the area of the portion of the sphere

r is the radius of the sphere

The SI unit of solid angle is the steradian (sr).

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 sphere of radius r has a total surface area of 4πr², so the solid angle subtended by the entire sphere at its center is: Ω=4πr²/r² =4π sr

Question-Answer on Supplementary Units

Conceptual Questions

1. What is a plane angle?
   • A plane angle is the angle subtended at the center of a circle by an arc of the circle. It is measured in radians.
2. 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.
3. How many steradians are there in a complete sphere?
   • A complete sphere subtends a solid angle of 4π 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 × 10⁸ m?

Solution:

Solid angle (Ω) = Area of the disc of the moon/(moon – earth distance)²
Ω = π(1 .737×10³)²/(3.84×10⁵)² = 2.045 x10^-5 sr

Multiple Choice Questions (MCQs)

1. The SI unit of plane angle is:

• (a) Degree
• (b) Radian
• (c) Steradian
• (d) None of these
• Answer: (b) Radian
• Explanation: The plane angle is measured in radians, where 1 radian is the angle subtended at the center of a circle by an arc equal to the radius.

2. The SI unit of solid angle is:

• (a) Degree
• (b) Radian
• (c) Steradian
• (d) None of these
• Answer: (c) Steradian
• Explanation: Solid angle is measured in steradians, and it represents the area subtended by an object on a sphere.

Test Your Knowledge(Quiz)

Do You Know?

• The total solid angle around a point in space is 4π steradians.
• 1 radian ≈ 57.3 degrees.
• The concept of radian and steradian helps in understanding wave optics, astronomy, and electromagnetic field theory.

Worksheet

1. Define radian and steradian.
2. Convert 2 radians into degrees.
3. What is the solid angle subtended by the Sun at the Earth, given that the Sun’s diameter is 1.39 × 10⁹ m and its distance from the Earth is 1.496 × 10¹¹ m?
4. Calculate the total solid angle subtended by a hemisphere.

Test Paper

Part A: Conceptual Questions

1. What is the difference between plane angle and solid angle?
2. How many radians are there in 90°?
3. Why is the unit steradian dimensionless?

Part B: Numerical Questions

1. The radius of a circular disc is 10 cm. Find the angle subtended by a 5 cm arc at the center.
2. The surface area of a sphere is 1256 cm². 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 = 180°/π ≈ 57.3°.
• 1 steradian is the angle subtended by an area equal to r² on a sphere.
• The total solid angle around a point is 4π sr.

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