Dimensional Formulae and Dimensional Equations
Once you can derive dimensions from defining equations, the next skill is recall speed: NTA expects the standard dimensional formulae of roughly thirty quantities to be retrievable in seconds. This page provides the master catalogue — mechanics, rotation, heat and electromagnetism — organised into families so memory holds, plus the formal distinction between a dimensional formula and a dimensional equation that NCERT draws and exams quote. Every row below has been cross-checked against its defining equation; where two quantities share a formula, the trap column tells you how NTA exploits it.
What are Dimensional Formulae and Equations? — Complete Theory
1. Formula vs equation: the NCERT distinction
A dimensional formula is the expression of a quantity as powers of base dimensions, e.g. [M L T⁻²] for force. A dimensional equation is that formula equated to the physical quantity: [F] = [M L T⁻²]. The distinction matters in "identify the dimensional equation" MCQs — the equation carries the quantity symbol on the left; the formula is just the right-hand side. NCERT's phrasing: "the dimensional formula is defined as the expression of the physical quantity in terms of its base unit's dimensions; the dimensional equation is obtained when we equate a physical quantity with its dimensional formula."
2. Master table — mechanics (M, L, T only)
| Quantity (defining hook) | Dimensional formula | SI unit | Memory hook / trap |
|---|---|---|---|
| Area (l × b) | [L²] | m² | no T at all |
| Volume | [L³] | m³ | density divides by this |
| Velocity (ds/dt) | [L T⁻¹] | m s⁻¹ | speed same formula |
| Acceleration (dv/dt) | [L T⁻²] | m s⁻² | g lives here |
| Force (ma) | [M L T⁻²] | N | 1 N = 10⁵ dyne |
| Momentum / Impulse (mv / FΔt) | [M L T⁻¹] | kg m s⁻¹ / N s | force × time removes one T |
| Work / Energy / Torque (F·s) | [M L² T⁻²] | J / N m | scalar work vs vector torque |
| Power (W/t) | [M L² T⁻³] | W | energy loses one T |
| Pressure / Stress / Young's modulus / Energy density (F/A) | [M L⁻¹ T⁻²] | Pa | force divided by area |
| Surface tension / Spring constant (F/l, F/x) | [M T⁻²] | N m⁻¹ | no L — trap pair |
| Viscosity (η = F·x/(A·v)) | [M L⁻¹ T⁻¹] | Pa s | 1 Pa s = 10 poise |
| Frequency / Angular velocity (1/T, dθ/dt) | [T⁻¹] | Hz / rad s⁻¹ | angle dimensionless |
| Gravitational constant G | [M⁻¹ L³ T⁻²] | N m² kg⁻² | only common M⁻¹ in mechanics |
| Planck constant h / Angular momentum (mvr) | [M L² T⁻¹] | J s | power loses one T from this |
| Moment of inertia (Σmr²) | [M L²] | kg m² | axis must be quoted |
| Latent heat (Q/m) | [L² T⁻²] | J kg⁻¹ | (velocity)² in disguise |
| Specific heat (Q/(mΔT)) | [L² T⁻² K⁻¹] | J kg⁻¹ K⁻¹ | heat page adds K⁻¹ |
| Pressure gradient (ΔP/Δx) | [M L⁻² T⁻²] | Pa m⁻¹ | one extra L⁻¹ over pressure |
3. Master table — heat & thermodynamics (+K)
| Quantity (defining hook) | Dimensional formula | SI unit | Memory hook / trap |
|---|---|---|---|
| Heat / Energy | [M L² T⁻²] | J | same as work |
| Specific heat capacity | [L² T⁻² K⁻¹] | J kg⁻¹ K⁻¹ | no M (divided by m) |
| Molar heat capacity | [M L² T⁻² K⁻¹ mol⁻¹] | J mol⁻¹ K⁻¹ | keeps M, adds mol⁻¹ |
| Gas constant R / Boltzmann kB / Entropy | [M L² T⁻² K⁻¹] (R adds mol⁻¹) | J mol⁻¹ K⁻¹ / J K⁻¹ | kB = R/NA |
| Thermal conductivity (Q·d/(A·ΔT·t)) | [M L T⁻³ K⁻¹] | W m⁻¹ K⁻¹ | power × length/(area × ΔT) |
| Stefan constant σ (power/(area·T⁴)) | [M T⁻³ K⁻⁴] | W m⁻² K⁻⁴ | no L in numerator route |
4. Master table — electricity & magnetism (+A)
| Quantity (defining hook) | Dimensional formula | SI unit | Memory hook / trap |
|---|---|---|---|
| Electric charge (It) | [A T] | C | current is the base quantity |
| Current density (I/A) | [A L⁻²] | A m⁻² | — |
| Potential / EMF (W/q) | [M L² T⁻³ A⁻¹] | V | power per current |
| Resistance (V/I) | [M L² T⁻³ A⁻²] | Ω | potential loses one A |
| Resistivity (RA/l) | [M L³ T⁻³ A⁻²] | Ω m | resistance + one L |
| Capacitance (q/V) | [M⁻¹ L⁻² T⁴ A²] | F | memorise — appears as-is |
| Magnetic field B (F/(qv)) | [M T⁻² A⁻¹] | T | no L — classic trap |
| Magnetic flux (BA) | [M L² T⁻² A⁻¹] | Wb | B picks up L² |
| Inductance (ε/(dI/dt)) | [M L² T⁻² A⁻²] | H | flux per current |
| Electric field E (F/q) | [M L T⁻³ A⁻¹] | V m⁻¹ / N C⁻¹ | force per charge |
Figure 1.5 — Dimension families at a glance
Each family bar lists its member quantities with the shared dimensional formula; scan for your target, then read the formula off the badge.
Exam read-out: every electricity formula is its mechanics ancestor with T-powers traded for A-powers; every heat formula is its ancestor wearing a K badge. Learn the mechanics column perfectly and the other two families follow.
Visualising Dimensional Formulae and Equations
Read every dimensional formula as a three-line invoice: how much mass, how much length, how much time. The family figure shows that invoices cluster: the "energy invoice" [M L² T⁻²] appears in mechanics (work), heat (heat itself) and electricity (potential × charge). When NTA asks you to match quantities to formulae, match invoices, not names — the moment you spot [M L² T⁻³ A⁻¹], think "power per ampere = volt" and the whole option column collapses.
Solved Examples on Dimensional Formulae and Equations (Step-by-Step)
Solved Example 1 — Capacitance from first principles
Q. Derive the dimensional formula of capacitance C = q/V.
Step 1. [q] = [A T] (charge = current × time). [V] = [W/q] = [M L² T⁻²]/[A T] = [M L² T⁻³ A⁻¹].
Step 2. [C] = [A T]/[M L² T⁻³ A⁻¹] = [A T] × [M⁻¹ L⁻² T³ A¹].
Step 3. Simplify exponents: M⁻¹, L⁻², T: T¹ × T³ = T⁴, A: 1 + 1 = 2. So [C] = [M⁻¹ L⁻² T⁴ A²], unit farad. ✔
ANSWER: [C] = [M⁻¹ L⁻² T⁴ A²]Solved Example 2 — Energy density of an electric field (JEE level)
Q. Show that (1/2)ε₀E² has the dimensions of energy density.
Step 1. From Coulomb's law F = q₁q₂/(4ε₀r²): [ε₀] = [q²]/[F·r²] = [A²T²]/([M L T⁻²][L²]) = [M⁻¹ L⁻³ T⁴ A²].
Step 2. [E] = [F/q] = [M L T⁻²]/[A T] = [M L T⁻³ A⁻¹], so [E²] = [M² L² T⁻⁶ A⁻²].
Step 3. [ε₀E²] = [M⁻¹ L⁻³ T⁴ A²][M² L² T⁻⁶ A⁻²] = [M¹ L⁻¹ T⁻² A⁰] = [M L⁻¹ T⁻²]. The factor ½ is a pure number and does not affect dimensions. ✔ Energy density [energy/volume] = [M L² T⁻²]/[L³] = [M L⁻¹ T⁻²]. Match confirmed.
ANSWER: [ε₀E²] = [M L⁻¹ T⁻²] = energy density ✔ (the ½ is dimensionless)Practice Questions on Dimensional Formulae and Equations (With Solutions)
Practice Q1 (Numerical-value type)
Q. For Planck's constant h with dimensional formula [Mᵃ L² T⁻¹], the sum a + 2 + (−1) equals?
Solution. [h] = E/ν = [M L² T⁻²]/[T⁻¹] = [M L² T⁻¹] → a = 1. Sum = 1 + 2 − 1 = 2.
ANSWER: 2Practice Q2 (Single-correct MCQ)
Q. The dimensional formula of surface tension is: (a) [M L⁻¹ T⁻²] (b) [M T⁻²] (c) [M L T⁻²] (d) [M L² T⁻²]
Solution. Surface tension = force per length = [M L T⁻²]/[L] = [M T⁻²]. Option (a) is pressure — the intended distractor.
ANSWER: (b) [M T⁻²]Practice Q3 (Assertion–Reason)
Q. Assertion (A): Planck's constant and angular momentum have the same dimensional formula. Reason (R): Both equal energy divided by frequency.
Solution. A is true: [h] = [L] = [M L² T⁻¹]. R is false: angular momentum is mvr (or Iω), not energy/frequency; the two merely share dimensions. A true, R false.
ANSWER: (c) A is true but R is falseKey Formulas & Takeaways
| Item | Statement | Exam use |
|---|---|---|
| Dimensional formula | powers-of-base-dimensions expression | match-the-column answers |
| Dimensional equation | [Quantity] = formula, with the symbol on the left | "identify the equation" MCQ |
| Must-memorise trio | [C] = [M⁻¹L⁻²T⁴A²] · [ε₀] = [M⁻¹L⁻³T⁴A²] · [B] = [M T⁻²A⁻¹] | JEE-level dimension questions |
| Family rule | heat = mechanics + K; EM = mechanics with A trading T | fast recall of 30 formulae |
| Collision list | work/torque · h/L · surface tension/spring constant · latent heat/v² | same-dimensions MCQs |
| Verify habit | re-derive any doubtful row from its defining equation | zero-error guarantee |
FAQs on Dimensional Formulae and Equations
What is a dimensional formula?
An expression showing a quantity in terms of base-dimension powers, e.g. [Pressure] = [M L⁻¹ T⁻²]; when it is set equal to the quantity symbol, [P] = [M L⁻¹ T⁻²], it is called a dimensional equation.
How do I find the dimensional formula of an unfamiliar quantity?
Write a defining equation with familiar quantities, replace each by its dimensions, and simplify the powers — e.g. capacitance C = q/V gives [M⁻¹ L⁻² T⁴ A²].
Which quantities have [M L² T⁻¹]?
Planck's constant (E = hν) and angular momentum (L = mvr); both reduce to [M L² T⁻¹], a favourite NTA match-the-dimension pair.
Is the dimensional formula unique to one quantity?
No — many quantities share a formula (work, torque, energy all [M L² T⁻²]); the formula identifies the 'family' of the quantity, not the quantity itself.
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