The trap that costs you marks here: getting the time exponent wrong by one. A student writes the dimensions of energy as [M L² T⁻¹] instead of [M L² T⁻²], or writes force as [M L T⁻¹] instead of [M L T⁻²]. One wrong exponent, one lost mark. NEET has tested this pattern repeatedly from 2021 through 2024.
What dimensions actually are. Every physical quantity can be expressed as a product of powers of the seven SI base quantities: mass (M), length (L), time (T), electric current (A), thermodynamic temperature (K), amount of substance (mol), and luminous intensity (cd). The dimensional formula of a quantity is this power-product written in square brackets. For example, force = mass × acceleration = M × L T⁻² = [M L T⁻²]. This is defined in NCERT Class 11 Physics Chapter 1, page 7.
The radian/steradian confusion. Plane angle (radian) and solid angle (steradian) are ratios — arc length divided by radius, or surface area divided by radius squared. They are dimensionless: [M⁰ L⁰ T⁰]. The fact that they have named units does not give them dimensions. NEET 2022 tested exactly this distinction.
How NEET tests dimensions. The common pattern: you are given a derived or unfamiliar combination of quantities and asked to find its dimensional formula, or you are given a dimensional formula like [M L T⁻² A⁻²] and asked which physical quantity it represents. The approach is mechanical — write the dimensional formula of each constituent, combine using algebra, and match. The error that loses marks is sloppy exponent arithmetic, especially on the time dimension.
Watch-out. When a formula has multiple terms added together (like F = αt² + βt), each term must have the same dimensions as F. Use this dimensional homogeneity to find the dimensions of the unknown constants α and β. Do not assume — derive.
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 the correct dimensional formula for energy?
Answer: C. Energy = work = force × displacement = [M L T⁻²] × [L] = [M L² T⁻²]. This is the standard dimensional formula for energy (NCERT Class 11 Physics Chapter 1, page 7).
Why A is wrong: A has the time exponent wrong by one: T⁻¹ instead of T⁻². This is the classic off-by-one trap on the time dimension.
Why B is wrong: B is the dimensional formula for force, not energy. Energy has an extra factor of length from the displacement.
Why D is wrong: D has M², which would require squaring mass — no standard energy formula does this.
The dimensional formula [M L T⁻² A⁻²] represents which physical quantity?
Answer: A. Permeability μ₀ has SI unit H/m = kg·m·s⁻²·A⁻². Writing in dimensional form: [M L T⁻² A⁻²]. Verify: force between two parallel currents involves μ₀, and the dimensions check out (NCERT Class 11 Physics Chapter 1).
Why B is wrong: B: Electric potential (volt) = [M L² T⁻³ A⁻¹]. The exponents on L, T, and A all differ from the given formula.
Why C is wrong: C: Electric resistance (ohm) = [M L² T⁻³ A⁻²]. The L exponent is 2 and T exponent is −3, not matching the given [M L¹ T⁻²].
Why D is wrong: D: Capacitance (farad) = [M⁻¹ L⁻² T⁴ A²]. The signs on every exponent are opposite to the given formula.
Plane angle and solid angle are:
Answer: B. Plane angle (radian) = arc length / radius = [L]/[L] = dimensionless. Solid angle (steradian) = surface area / radius² = [L²]/[L²] = dimensionless. Both are pure ratios with no dimensions, even though they carry named SI units (NCERT Class 11 Physics Chapter 1).
Why A is wrong: A is wrong because neither angle has dimensions. Having a unit name (radian, steradian) does not imply having dimensions — this is a common confusion (trap: treating radian/steradian as dimensional because they have unit names).
Why C is wrong: C incorrectly assigns [L] to plane angle and [L²] to solid angle by not cancelling the denominator in the ratio definitions (arc/radius and area/r²).
Why D is wrong: D splits the two angles arbitrarily. Both are defined as ratios of like-dimensioned quantities, so both are dimensionless.
If force F is given by F = αt² + βt, where t is time and F has dimensions [M L T⁻²], what are the dimensions of α?
Answer: B. By dimensional homogeneity, the term αt² must have the same dimensions as F. So [α] × [T²] = [M L T⁻²], giving [α] = [M L T⁻²] / [T²] = [M L T⁻⁴] (NCERT Class 11 Physics Chapter 1, page 7 — principle of dimensional homogeneity).
Why A is wrong: A results from dividing by T¹ instead of T². This is the off-by-one error on the time exponent — α multiplies t², not t.
Why C is wrong: C results from dividing by T⁻¹ (i.e., multiplying by T), which reverses the operation entirely.
Why D is wrong: D has L² instead of L¹. The length exponent in force is 1, not 2 — confusing energy [M L² T⁻²] with force [M L T⁻²].
The dimensions of the ratio (energy / gravitational constant) are:
Answer: A. Energy E = [M L² T⁻²]. Gravitational constant G: from F = Gm₁m₂/r², G = Fr²/(m₁m₂) = [M L T⁻²][L²]/[M²] = [M⁻¹ L³ T⁻²]. So E/G = [M L² T⁻²] / [M⁻¹ L³ T⁻²] = [M² L⁻¹ T⁰] = [M² L⁻¹]. The T⁻² cancels completely.
Why B is wrong: B retains T⁻² in the result, failing to cancel T⁻² in the numerator with T⁻² in the denominator. This is the most common arithmetic slip in this type of question.
Why C is wrong: C has T⁻¹, which would require an odd mismatch in time exponents — there is no source of a single T⁻¹ in this division.
Why D is wrong: D has L¹ instead of L⁻¹. This comes from subtracting the length exponents in the wrong order (2 − 3 = −1, not +1).
If velocity [v], force [F], and time [T] are chosen as fundamental quantities, the dimensional formula of mass in this new system is:
Answer: D. D is correct. In SI, F = [M L T⁻²] and v = [L T⁻¹]. From F = ma, mass = F/a, and dimensionally a = v/T, so M = F·T/v = [F T v⁻¹]. Check: [M L T⁻²]·[T]·[L T⁻¹]⁻¹ = [M L T⁻²]·[T]·[L⁻¹ T] = [M L T⁻² L⁻¹ T²] = [M]. Confirmed.
Why A is wrong: A is wrong because F·v·T = [M L T⁻²]·[L T⁻¹]·[T] = [M L² T⁻²], which is the dimensional formula of energy, not mass. Velocity has to divide, not multiply: acceleration is v/T, and mass is force divided by acceleration.
Why B is wrong: B uses T² instead of T. F·T² = [M L T⁻²]·[T²] = [M L], which has a leftover length dimension.
Why C is wrong: C omits the time factor. F/v = [M L T⁻²]/[L T⁻¹] = [M T⁻¹], which is not pure mass — it still carries T⁻¹.
Which of the following quantities is dimensionless?
Answer: C. Strain = change in length / original length = [L]/[L] = dimensionless. It is a pure ratio, similar to how radian is defined (NCERT Class 11 Physics Chapter 1).
Why A is wrong: A: Angular velocity = angle/time = [T⁻¹]. Although the angle (radian) is dimensionless, dividing by time introduces a time dimension.
Why B is wrong: B: Torque = force × distance = [M L² T⁻²]. It has the same dimensions as energy (but is a different physical concept — a vector vs a scalar).
Why D is wrong: D: Pressure = force/area = [M L⁻¹ T⁻²]. It clearly has dimensions.
In the equation F = αt² + βt, where F is force [M L T⁻²] and t is time, the dimensions of β are:
Answer: D. By dimensional homogeneity, βt must have the same dimensions as F. So [β] × [T] = [M L T⁻²], giving [β] = [M L T⁻²] / [T] = [M L T⁻³].
Why A is wrong: A gives the dimensions of F itself, not β. This would mean β is dimensionless, but β multiplies t, so it must carry dimensions that compensate for [T].
Why B is wrong: B is the dimensions of α (from αt²), not β. Confusing which constant goes with which power of t is a common error.
Why C is wrong: C has L² instead of L¹. The length exponent in force is 1, not 2. This error comes from confusing the dimensional formula of force with that of energy.
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Pattern: Deduce the dimensions of a derived combination of quantities (NEET 2022, 2024 pattern — dimensions of a derived quantity).
Given
A physical quantity P is defined as P = E²/(G · h), where:• E is energy• G is the universal gravitational constant• h is Planck's constant
Find the dimensions of P.
Required
The dimensional formula of P = E²/(G · h).
Concept
Write the dimensional formula of each constituent quantity, then combine using the rules of exponents. The principle of dimensional homogeneity guarantees a unique result.
Formula
• E (energy) = [M L² T⁻²]• G (gravitational constant): from F = Gm₁m₂/r², G = [M⁻¹ L³ T⁻²]• h (Planck's constant): from E = hν, h = E/ν = [M L² T⁻²]/[T⁻¹] = [M L² T⁻¹]
Substitution
P = E² / (G · h)
= [M L² T⁻²]² / ([M⁻¹ L³ T⁻²] · [M L² T⁻¹])
Numerator: [M² L⁴ T⁻⁴]
Denominator: [M⁻¹ L³ T⁻²] · [M L² T⁻¹]
= [M⁻¹⁺¹ L³⁺² T⁻²⁻¹]
= [M⁰ L⁵ T⁻³]
Calculation
P = [M² L⁴ T⁻⁴] / [M⁰ L⁵ T⁻³]
= [M²⁻⁰ L⁴⁻⁵ T⁻⁴⁻(⁻³)]
= [M² L⁻¹ T⁻¹]
Cross-check the time exponent carefully: −4 − (−3) = −4 + 3 = −1. This is the step where the off-by-one trap strikes — students who rush write T⁻² or T⁰.
Note: the exponents 2 (on E) and 1 (on G and h) are exact mathematical powers defined by the formula P = E²/(Gh). They are not measurements and do not contribute to any error or significant-figure analysis.
Final answer
P = [M² L⁻¹ T⁻¹]
Common trap
The time-exponent off-by-one error. In the denominator, T⁻² × T⁻¹ = T⁻³ (not T⁻²). Then −4 − (−3) = −1 (not −2). Students who skip explicit exponent arithmetic and guess get T⁻² in the final answer, which is wrong.
Similar NEET-style question
Find the dimensions of the quantity σ = F²/(E · v), where F is force, E is energy, and v is velocity. (Answer: [M L⁻¹ T⁻¹] — verify by writing each dimensional formula and combining: F² = M²L²T⁻⁴ and E·v = ML³T⁻³, so the quotient is M L⁻¹ T⁻¹, the dimensions of viscosity.)
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The dimensions of a physical quantity are the powers (exponents) to which the base quantities are raised to represent that quantity. The bracket [Q] denotes the dimensions of Q. For example, [velocity] = [L T⁻¹]; [force] = [M L T⁻²].
-- NCERT Class 11 Physics, Ch. 1, p. 7The 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 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.
More in Units and Measurements: 7 exam traps and mistakes · 7 question patterns from its other lessons.
The quantities which have the same dimensions as those of solid angle are :
The dimensions [MLT⁻²A⁻²] belong to the
miscounts power of T
Off-by-one on time exponent (e.g. -1 vs -2)
treats radian as dimensional
Confuses 'has unit' with 'has dimensions'
Page numbers are the ones printed in the current NCERT textbook (2023 rationalised edition), unless marked pre-2023. The books are free at ncert.nic.in.
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