The trap that costs marks on this topic: confusing units with dimensions. Radian and steradian have unit names — but they are dimensionless. NEET distractors exploit exactly this confusion.
Fundamental units are the seven SI base units — metre (length), kilogram (mass), second (time), ampere (electric current), kelvin (temperature), mole (amount of substance), and candela (luminous intensity). These are independently defined and cannot be expressed in terms of each other (NCERT Class 11 Physics Chapter 1, page 1).
Two supplementary units — radian (plane angle) and steradian (solid angle) — were historically listed alongside the base units. After the 1995 CGPM decision, they are classified as derived, dimensionless units. A radian is arc-length divided by radius (length/length = dimensionless). A steradian is surface-area divided by radius-squared (length²/length² = dimensionless). They carry unit names purely for convenience — they have no dimensions.
Derived units are built from the fundamental units through multiplication, division, or exponentiation. Examples: newton (kg·m·s⁻²), joule (kg·m²·s⁻²), pascal (kg·m⁻¹·s⁻²), watt (kg·m²·s⁻³).
The distinction matters for NEET because:
Watch-out: "dimensionless" does not mean "unitless." Radian is dimensionless but has a unit name. This is the single distinction NEET tests on this topic.
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
How many fundamental (base) units are there in the International System of Units (SI)?
Which of the following is NOT an SI base unit?
Which of the following pairs are both dimensionless quantities?
The SI unit of force is the newton. Express 1 newton in terms of fundamental SI units.
The SI unit of pressure is the pascal. Which of the following correctly expresses 1 pascal in terms of base SI units?
Which of the following statements about radian and steradian is correct?
A student claims: "Since radian and steradian have different unit symbols (rad and sr), they must have different dimensions." What is wrong with this claim?
The energy stored in a capacitor is given by E = ½CV², where C is capacitance (unit: farad) and V is voltage (unit: volt). Express the farad in terms of fundamental SI units.
Pattern: Derive the SI base-unit expression for a derived quantity (aligned with the unit-vs-dimension conceptual theme of this lesson).
Given
The power dissipated in a resistor is P = V²/R, where V is voltage and R is resistance. - 1 watt (P) = 1 kg·m²·s⁻³ - 1 volt (V) = 1 kg·m²·s⁻³·A⁻¹
Required
Express the ohm (unit of resistance R) in terms of fundamental SI units.
Concept
From P = V²/R, we get R = V²/P. Since both watt and volt are already expressed in base units, we can derive the ohm by algebraic substitution.
Formula
R = V²/P (dimensional relationship)
Substitution
ohm = (kg·m²·s⁻³·A⁻¹)² / (kg·m²·s⁻³) Numerator (V²): kg²·m⁴·s⁻⁶·A⁻² Denominator (P): kg·m²·s⁻³
Calculation
ohm = kg²·m⁴·s⁻⁶·A⁻² / (kg·m²·s⁻³) Divide each base unit: - kg: 2 − 1 = 1 - m: 4 − 2 = 2 - s: −6 − (−3) = −3 - A: −2 − 0 = −2 ohm = kg·m²·s⁻³·A⁻² Note: the exponents in the given volt and watt expressions are exact definitions, not measured values — they do not carry uncertainty.
Final answer
**1 ohm = 1 kg·m²·s⁻³·A⁻²**
Common trap
The common error is mishandling the squaring step — students sometimes square only the numerical coefficient or forget to square *every* base-unit exponent in the volt expression. For instance, forgetting to square the A⁻¹ term yields A⁻¹ instead of A⁻² in the final answer.
Similar NEET-style question
"Express the henry (unit of inductance) in terms of fundamental SI units, given that the energy stored in an inductor is E = ½LI²." ---
Although the number of physical quantities is large, only a limited number of units is needed because all physical quantities are inter-related. The chosen quantities are called fundamental (base) quantities; the rest are derived from them.
-- NCERT Class 11 Physics, Ch. 1, p. 1The 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
Test yourself on this topic with real past-paper questions:
Practice this topic →Get a structured 30-day Mechanics plan and a complete formula booklet — delivered to your inbox instantly.