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The quantities which have the same dimensions as those of solid angle are:
Solid angle is dimensionless [M⁰L⁰T⁰]! Strain (ΔL/L) and Plane angle (Arc/Radius) are BOTH dimensionless!
Solid angle Ω subtended by an area element dA at distance r is defined as the ratio of area to the square of distance:
Strain ε is defined as fractional change in dimension (e.g. change in length per unit original length):
Plane angle θ is defined as the ratio of arc length to radius:
Evaluating the other quantities present in distractors: - Arc: [L] - Stress: [Force / Area] = [M L⁻¹ T⁻²] - Angular speed: [Angle / Time] = [T⁻¹] Only the pair "strain and angle" are both dimensionless [M⁰L⁰T⁰].
Diagnose every choice: Know exactly why the right option works and why distractors fail.
Arc has dimension of length [L].
Angular speed is [T⁻¹] and stress is [M L⁻¹ T⁻²].
Both are dimensionless ratios [M⁰L⁰T⁰], matching solid angle.
Stress has dimensions of pressure [M L⁻¹ T⁻²].
Students often believe that because solid angle has a unit (steradian) and angle has a unit (radian), they must have physical dimensions.
"Physical quantities defined as the ratio of two like quantities (such as strain, plane angle, and solid angle) possess no physical dimensions ([M⁰L⁰T⁰])."
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