The trap that costs marks on SI Units questions is simpler than you think: confusing a unit name with a dimension. Radian and steradian both have names — they appear in tables, they have abbreviations — but they are dimensionless. They are ratios: arc length divided by radius (radian), surface area divided by radius squared (steradian). NEET distractors exploit this by offering "radian has dimension of length" or "steradian has dimension of area."
The SI system, as defined by the CGPM (NCERT Class 11 Physics Chapter 1, page 2), rests on seven base quantities with seven base units: length (metre), mass (kilogram), time (second), electric current (ampere), thermodynamic temperature (kelvin), amount of substance (mole), and luminous intensity (candela). Every other unit — newton, joule, pascal, watt — is derived from these seven by multiplication, division, or exponentiation. No additional independent quantities are needed.
Two supplementary quantities — plane angle and solid angle — were historically listed alongside the base quantities but were reclassified as dimensionless derived quantities. Their units (radian and steradian) remain in use for clarity, but they carry no independent dimension. This is where the confusion sits: having a unit name does not mean having a dimension.
For NEET, anchor on these facts:
Watch out for distractors that list radian or steradian as "having dimensions" or that claim there are nine base quantities. There are seven.
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
Which of the following is NOT an SI base unit?
The SI unit of amount of substance is:
How many base quantities are there in the International System of Units (SI)?
Plane angle (radian) and solid angle (steradian) are:
The SI unit of luminous intensity is:
Which of the following statements about SI base units is correct?
The unit 'pascal' is equivalent to which of the following in terms of SI base units?
A student claims that the radian has the dimensional formula [L]. Which of the following correctly identifies the error?
Pattern: Plane vs solid angle dimensions — identifying dimensionless quantities (pattern anchored to NEET 2022)
Given
A physics teacher writes the following on the board: - Plane angle θ = arc length / radius - Solid angle Ω = surface area / (radius)² A student argues that since arc length has the dimension [L] and surface area has the dimension [L²], the plane angle must have dimension [L] and the solid angle must have dimension [L²].
Required
Determine the correct dimensions of plane angle and solid angle. Identify the student's error.
Concept
A quantity's dimension depends on the complete defining expression, not just the numerator. When a quantity is defined as a ratio of two quantities with the same dimension, the dimensions cancel, yielding a dimensionless result.
Formula
Plane angle: θ = s / r, where both s and r have dimension [L]. Solid angle: Ω = A / r², where A has dimension [L²] and r² has dimension [L²].
Substitution
Plane angle: [θ] = [L] / [L] = [M⁰L⁰T⁰] Solid angle: [Ω] = [L²] / [L²] = [M⁰L⁰T⁰]
Calculation
Both ratios yield [M⁰L⁰T⁰] — no surviving dimension of length in either case. No arithmetic beyond the dimensional cancellation is needed.
Final answer
Both plane angle and solid angle are **dimensionless** quantities. Their dimensions are [M⁰L⁰T⁰]. They have SI unit names (radian and steradian) for practical clarity, but these names do not confer dimensions. The student's error: treating only the numerator's dimension as the quantity's dimension, while ignoring the denominator, which has the same dimension and cancels it completely.
Common trap
Confusing "has a unit name" with "has a dimension." Many students assume that because radian and steradian appear in tables of units — just like metre and kilogram do — they must carry independent dimensions. They do not. A unit name is a label for convenience; a dimension is a fundamental independent physical attribute. Radian and steradian are ratios of like quantities.
Similar NEET-style question
"Which of the following pairs of physical quantities are both dimensionless? (A) Strain and refractive index (B) Force and impulse (C) Angle and angular velocity (D) Solid angle and torque" Answer: (A). Strain = ΔL/L (dimensionless ratio); refractive index = c/v (dimensionless ratio). This tests the same principle: ratios of like-dimensioned quantities yield dimensionless results. ---
The SI is the internationally accepted system of units, based on seven base units: metre (length), kilogram (mass), second (time), ampere (electric current), kelvin (thermodynamic temperature), mole (amount of substance) and candela (luminous intensity).
-- NCERT Class 11 Physics, Ch. 1, p. 2Table of seven SI base units with symbols: m (metre), kg (kilogram), s (second), A (ampere), K (kelvin), mol (mole), cd (candela). Derived units are formed from these (e.g. N for newton = kg·m·s⁻², J for joule = N·m, W for watt = J·s⁻¹).
-- NCERT Class 11 Physics, Ch. 1, p. 2The maximum relative error in a power expression is the sum of the absolute exponents weighted by the relative errors of the bases. Negative exponents (divisions) still take the |.| value because we want the worst-case error.
| Symbol | Quantity | SI Unit |
|---|---|---|
| Z | Result | (combined) |
| p, q, r | Exponents (signed) | (dimensionless) |
| A, B, C | Measured quantities | (measured) |
When two measured quantities are multiplied or divided, the maximum RELATIVE errors add. The absolute error in the result is then Delta_Z = Z * (relative-error sum).
| Symbol | Quantity | SI Unit |
|---|---|---|
| Z | Result of product/quotient | (combined unit) |
| A | First measured quantity | (measured) |
| B | Second measured quantity | (measured) |
| Delta_A/A | Relative error in A | (dimensionless) |
| Delta_B/B | Relative error in B | (dimensionless) |
When two quantities are added or subtracted, the maximum absolute errors of the inputs simply add to give the maximum absolute error of the output. The relative error is NOT what adds in this case.
| Symbol | Quantity | SI Unit |
|---|---|---|
| Z | Result of sum/difference | (same as A,B) |
| A | First measured quantity | (measured) |
| B | Second measured quantity | (measured) |
| Delta_Z | Maximum absolute error in Z | (same as A,B) |
| Delta_A | Maximum absolute error in A | (same as A) |
| Delta_B | Maximum absolute error in B | (same as B) |
These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.
Category: Similar Terms
Student gets the time exponent wrong by 1 (e.g. T⁻¹ vs T⁻²) when manipulating dimensional formulas.
Question asks for dimensions of a derived combination (e.g. E/G, F = αt² + βt) where time exponent matters.
Write each base quantity's dimensional formula explicitly, then combine. Common errors: dividing forces forgets sub of T exponents; energy/length includes implicit time. Always check final units against expected SI.
Category: Similar Terms
Student sums relative errors of all measured quantities without weighting by the exponent. For ρ = m/(πr²L), the relative error contribution of r is 2 × Δr/r, NOT Δr/r — the exponent of r in the formula carries through as a multiplicative factor.
Question gives a derived quantity formula with mixed-power dependencies; asks for the max relative error. Distractors omit the power factor.
Always write the full general rule: for Z = A^p B^q C^r, ΔZ/Z = |p|·ΔA/A + |q|·ΔB/B + |r|·ΔC/C. Identify the powers (1, 2, 3, ½) before adding.
Category: Similar Terms
Student conflates random errors (statistical, unpredictable, reduced by averaging) with instrumental errors (consistent bias from the apparatus) or with systematic errors (consistent bias from the method). Each has a distinct definition and different mitigation.
Question describes an error source and asks for its taxonomic category. Distractors include cognate categories.
Memorise the 5-category taxonomy: PERSONAL (observer-side), INSTRUMENTAL (apparatus calibration), LEAST-COUNT (instrument resolution floor), RANDOM (statistical, reduced by repeated trials), SYSTEMATIC (method-level bias, NOT reduced by averaging).
Category: Similar Terms
Student swaps which is the input vs the output: least count = pitch / N (where N is the number of circular-scale divisions). Distractors offer the ratio inverted or the wrong unit.
Question gives one of (pitch, N, least count) and asks for another; distractors offer the inverted ratio or off-by-factor-of-10.
Anchor on the definition: least count is the SMALLEST measurement the instrument can resolve. It is always SMALLER than the pitch. So pitch = LC × N (and not LC = pitch × N).
Category: Similar Terms
Student applies the 'fewest significant figures' rule (which governs multiplication and division) to a sum or difference. Subtraction of two measured numbers must instead reflect the FEWEST decimal places.
Question involves addition/subtraction of measured numbers with very different magnitudes or decimal-place counts (e.g. 9.99 - 0.0099). Distractors offer answers rounded by sig-fig rule rather than decimal-place rule.
Memorise: multiplication/division → fewest SIGNIFICANT FIGURES; addition/subtraction → fewest DECIMAL PLACES. Always identify which arithmetic operation is being performed before applying any rule.
Category: Similar Terms
Student treats radian/steradian as having dimensions because they have unit names.
Question asks about dimensions of plane angle, solid angle, or comparison.
Plane angle (radian) and solid angle (steradian) are DIMENSIONLESS — they're ratios (arc/radius for radian; surface-area/r² for steradian). They have unit NAMES for clarity but no dimensions.
Category: Similar Terms
Confusing whether N or N+1 is the smaller count when (N+1) divisions of vernier match N divisions of main scale.
Question gives '(N+1) divisions of vernier coincide with N divisions of main' or similar phrasing.
Always interpret carefully: N+1 vernier divisions span the SAME LENGTH as N main divisions. So 1 VSD = (N/(N+1)) MSD; vernier constant = 1 MSD - 1 VSD = 1 MSD / (N+1). Result smaller than 1 MSD.
Root cause: formula misuse
Use Delta_Z = Delta_A + Delta_B for sums/differences (absolute errors add). Use Delta_Z/Z = Delta_A/A + Delta_B/B for products/quotients (relative errors add). They are NOT interchangeable — the rule is dictated by whether the operation is additive or multiplicative.
Distractor option uses the wrong rule (e.g. quotes a small relative error for a sum where absolute errors should add).
Root cause: concept gap
For multiplication/division, the result has the fewest significant figures of any input. For addition/subtraction, the result has the fewest decimal places of any input. Decimal places ≠ significant figures.
Distractor truncates a sum to too few significant figures by applying the multiplication rule.
The quantities which have the same dimensions as those of solid angle are :
The dimensions [MLT–2A–2] belong to the
Plane angle and solid angle have
miscounts power of T
Off-by-one on time exponent (e.g. -1 vs -2)
forgets power of two on radius
Default to summing all relative errors with weight 1
linear sum no powers
Adding all relative errors with weight 1
treats radian as dimensional
Confuses 'has unit' with 'has dimensions'
instrumental misclassified as random
Both feel 'unavoidable'; need to recognise instrumental ≠ random
swaps pitch and LC
Confusing which side of the formula is asked
no rounding
Calculator answer kept verbatim
applies mult rule to subtraction
Default to 'fewest sig figs' without distinguishing subtraction's decimal-places rule
swap N and N+1
Confusing which side has the larger count
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