Question

Plane angle and solid angle have:

• A. Units but no dimensions
• B. Dimensions but no units
• C. No units and no dimensions
• D. Both units and dimensions

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Dimensions describe the physical nature of a quantity (like Mass, Length, Time).
Units are the standard scales used to measure those quantities.
A dimensionless quantity is one that is defined as a ratio of two similar physical quantities.
Step 2: Detailed Explanation:
1. Plane Angle (\(\theta\)):
It is defined as the ratio of the arc length (\(s\)) to the radius (\(r\)).
\[ \theta = \frac{s}{r} \]
Since both \(s\) and \(r\) have dimensions of Length \([L]\), the angle is \([L]/[L] = [L^0]\). It has no dimensions.
However, it has a standard SI unit called the radian.

2. Solid Angle (\(\Omega\)):
It is defined as the ratio of the spherical area (\(A\)) to the square of the radius (\(r^2\)).
\[ \Omega = \frac{A}{r^2} \]
Since \(A\) has dimensions of \([L^2]\) and \(r^2\) also has \([L^2]\), the solid angle is \([L^2]/[L^2] = [L^0]\). It is dimensionless.
However, it has a standard SI unit called the steradian.
Step 3: Final Answer:
Both plane and solid angles are unique in that they possess units but are dimensionless.