Prerequisites
Before studying dimensional formulas, you should understand:
- SI Units — Base and derived units
What Are Dimensions?
Dimensions of a physical quantity are the powers (exponents) to which the base quantities are raised to represent that quantity.
The Seven Fundamental Dimensions
We use square brackets [ ] to denote dimensions:
| Base Quantity | Symbol | Dimension |
|---|---|---|
| Mass | M | [M] |
| Length | L | [L] |
| Time | T | [T] |
| Electric Current | I or A | [I] or [A] |
| Temperature | θ or K | [θ] or [K] |
| Amount of Substance | N or mol | [N] or [mol] |
| Luminous Intensity | J or cd | [J] or [cd] |
Dimensional Formula vs Unit
Don’t confuse these two concepts:
| Physical Quantity | Unit | Dimensional Formula |
|---|---|---|
| Velocity | m/s | [M⁰L¹T⁻¹] or [LT⁻¹] |
| Force | newton (N) | [MLT⁻²] |
| Energy | joule (J) | [ML²T⁻²] |
- Unit = actual measurement standard (meter, kilogram, etc.)
- Dimensional Formula = abstract relationship between base quantities
Writing Dimensional Formulas: Step-by-Step
Method 1: From Definition/Formula
Example: Velocity
$$v = \frac{\text{displacement}}{\text{time}} = \frac{L}{T}$$ $$\boxed{[v] = [L][T]^{-1} = [M^0 L^1 T^{-1}]}$$Example: Force
$$F = ma = M \times \frac{L}{T^2}$$ $$\boxed{[F] = [M][L][T]^{-2} = [M^1 L^1 T^{-2}]}$$Method 2: From SI Unit
Example: Pressure
Unit: Pascal (Pa) = N/m² = kg/(m·s²)
$$\boxed{[P] = \frac{[M]}{[L][T]^2} = [M^1 L^{-1} T^{-2}]}$$In dimensional formulas:
- Powers can be positive, negative, or zero
- Powers are always rational numbers
- We write [M⁰L¹T⁻¹] or simply [LT⁻¹] (omitting M⁰)
Interactive Demo: Visualize Dimensional Formulas
See how physical quantities are composed from fundamental dimensions.
Comprehensive Dimensional Formula Table
1. Mechanical Quantities
| Physical Quantity | Symbol | Formula/Definition | Dimensional Formula | SI Unit |
|---|---|---|---|---|
| Length | l, d, r | — | [L] | m |
| Mass | m | — | [M] | kg |
| Time | t | — | [T] | s |
| Area | A | l² | [L²] | m² |
| Volume | V | l³ | [L³] | m³ |
| Density | ρ | m/V | [ML⁻³] | kg/m³ |
| Velocity | v | s/t | [LT⁻¹] | m/s |
| Acceleration | a | v/t | [LT⁻²] | m/s² |
| Momentum | p | mv | [MLT⁻¹] | kg·m/s |
| Force | F | ma | [MLT⁻²] | N |
| Impulse | J | Ft | [MLT⁻¹] | N·s |
| Work/Energy | W, E | Fs | [ML²T⁻²] | J |
| Power | P | W/t | [ML²T⁻³] | W |
| Pressure | p | F/A | [ML⁻¹T⁻²] | Pa |
| Angular velocity | ω | θ/t | [T⁻¹] | rad/s |
| Angular acceleration | α | ω/t | [T⁻²] | rad/s² |
| Moment of inertia | I | mr² | [ML²] | kg·m² |
| Torque | τ | Fr | [ML²T⁻²] | N·m |
| Angular momentum | L | Iω | [ML²T⁻¹] | kg·m²/s |
Work, Energy, and Torque all have the same dimensions [ML²T⁻²], but:
- Work and Energy have the same unit: joule (J)
- Torque has unit: N·m (not joule!)
This is because torque is a vector product (r × F) while work is a scalar product (F · s).
2. Thermal Quantities
| Physical Quantity | Symbol | Dimensional Formula | SI Unit |
|---|---|---|---|
| Temperature | T, θ | [K] or [θ] | K |
| Heat | Q | [ML²T⁻²] | J |
| Specific heat | s, c | [L²T⁻²K⁻¹] | J/(kg·K) |
| Latent heat | L | [L²T⁻²] | J/kg |
| Thermal conductivity | k | [MLT⁻³K⁻¹] | W/(m·K) |
| Stefan’s constant | σ | [MT⁻³K⁻⁴] | W/(m²·K⁴) |
| Boltzmann constant | k_B | [ML²T⁻²K⁻¹] | J/K |
| Gas constant | R | [ML²T⁻²K⁻¹mol⁻¹] | J/(mol·K) |
3. Electricity and Magnetism
| Physical Quantity | Symbol | Dimensional Formula | SI Unit |
|---|---|---|---|
| Charge | q, Q | [IT] or [AT] | C |
| Current | I | [I] or [A] | A |
| Voltage/EMF | V, ε | [ML²T⁻³I⁻¹] | V |
| Resistance | R | [ML²T⁻³I⁻²] | Ω |
| Capacitance | C | [M⁻¹L⁻²T⁴I²] | F |
| Electric field | E | [MLT⁻³I⁻¹] | V/m or N/C |
| Magnetic field | B | [MT⁻²I⁻¹] | T |
| Magnetic flux | Φ | [ML²T⁻²I⁻¹] | Wb |
| Inductance | L | [ML²T⁻²I⁻²] | H |
| Permittivity | ε₀ | [M⁻¹L⁻³T⁴I²] | F/m |
| Permeability | μ₀ | [MLT⁻²I⁻²] | H/m |
4. Waves and Oscillations
| Physical Quantity | Symbol | Dimensional Formula | SI Unit |
|---|---|---|---|
| Frequency | f, ν | [T⁻¹] | Hz |
| Wavelength | λ | [L] | m |
| Wave velocity | v | [LT⁻¹] | m/s |
| Amplitude | A | [L] | m |
| Time period | T | [T] | s |
| Spring constant | k | [MT⁻²] | N/m |
| Surface tension | γ, σ | [MT⁻²] | N/m |
| Viscosity | η | [ML⁻¹T⁻¹] | Pa·s |
5. Modern Physics
| Physical Quantity | Symbol | Dimensional Formula | SI Unit |
|---|---|---|---|
| Planck’s constant | h | [ML²T⁻¹] | J·s |
| Work function | φ | [ML²T⁻²] | J or eV |
| Radioactive decay constant | λ | [T⁻¹] | s⁻¹ |
| Half-life | t₁/₂ | [T] | s |
6. Special Constants
| Physical Quantity | Symbol | Dimensional Formula | SI Unit |
|---|---|---|---|
| Gravitational constant | G | [M⁻¹L³T⁻²] | N·m²/kg² |
| Acceleration due to gravity | g | [LT⁻²] | m/s² |
| Universal gas constant | R | [ML²T⁻²K⁻¹mol⁻¹] | J/(mol·K) |
| Avogadro’s number | N_A | [mol⁻¹] | mol⁻¹ |
Dimensionless Quantities
Some quantities have no dimensions (all powers are zero):
| Quantity | Type | Example |
|---|---|---|
| Pure numbers | Counting | 1, 2, π, e |
| Ratios of same quantities | Relative | Strain = ΔL/L |
| Angles | Trigonometric | θ, sin θ, cos θ, tan θ |
| Relative density | Ratio | ρ_substance/ρ_water |
| Refractive index | Ratio | n = c/v |
| Coefficient of friction | Ratio | μ = F_friction/F_normal |
| Poisson’s ratio | Ratio | σ = -lateral strain/longitudinal strain |
| Reynold’s number | Dimensionless parameter | Re = ρvL/η |
Mistake: Thinking angles have dimensions because they’re measured in degrees or radians.
Fact: Angles are dimensionless! Radians are defined as arc length/radius = L/L = 1 (dimensionless).
Same for solid angle (steradians): area/radius² = L²/L² = 1.
Principle of Homogeneity of Dimensions
In any correct physical equation, the dimensions of all terms on both sides must be the same.
This is called the Principle of Homogeneity.
Example: Checking Homogeneity
$$s = ut + \frac{1}{2}at^2$$Check each term:
- [s] = [L]
- [ut] = [LT⁻¹][T] = [L] ✓
- [½at²] = [LT⁻²][T²] = [L] ✓
All terms have dimension [L], so the equation is dimensionally correct.
- Only terms with same dimensions can be added or subtracted
- Dimensionless constants (like ½, π, e) don’t affect dimensional analysis
- Arguments of exponential, logarithmic, and trigonometric functions must be dimensionless
Memory Tricks
Remember: “Power is Force times Velocity” → Powers add when multiplying!
All have [ML²T⁻²]:
- Work
- Energy (kinetic, potential, heat)
- Torque
- Moment of force
Memory: “WET Moments” = Work, Energy, Torque, Moments
- Pressure = Force/Area → [MLT⁻²][L⁻²] = [ML⁻¹T⁻²]
- Surface tension = Force/Length → [MLT⁻²][L⁻¹] = [MT⁻²]
- Stress = Force/Area → [ML⁻¹T⁻²]
Practice Problems
Level 1: JEE Main Basics
Find the dimensional formula of kinetic energy using KE = ½mv².
Solution:
$$[KE] = [m][v]^2 = [M][LT^{-1}]^2$$ $$= [M][L^2T^{-2}]$$ $$= \boxed{[ML^2T^{-2}]}$$What are the dimensions of universal gravitational constant G in the equation F = Gm₁m₂/r²?
Solution:
$$F = \frac{Gm_1m_2}{r^2}$$ $$G = \frac{Fr^2}{m_1m_2}$$ $$[G] = \frac{[MLT^{-2}][L^2]}{[M][M]} = \frac{[ML^3T^{-2}]}{[M^2]}$$ $$= \boxed{[M^{-1}L^3T^{-2}]}$$Level 2: JEE Main/Advanced
Check if the equation $v^2 = u^2 + 2as$ is dimensionally correct.
Solution: LHS: $[v^2] = [LT^{-1}]^2 = [L^2T^{-2}]$
RHS:
- $[u^2] = [L^2T^{-2}]$
- $[2as] = [LT^{-2}][L] = [L^2T^{-2}]$
Both terms on RHS have [L²T⁻²], and LHS = RHS.
Answer: ✓ Dimensionally correct!
Find the dimensions of a/b in the equation P = (a - t²)/bx, where P = pressure, t = time, x = distance.
Solution:
$$P = \frac{a - t^2}{bx}$$Since (a - t²) must have same dimensions as t²:
$$[a] = [t^2] = [T^2]$$From dimensional homogeneity:
$$[P] = \frac{[a]}{[b][x]} = \frac{[T^2]}{[b][L]}$$ $$[ML^{-1}T^{-2}] = \frac{[T^2]}{[b][L]}$$ $$[b] = \frac{[T^2]}{[ML^{-1}T^{-2}][L]} = \frac{[T^2]}{[MT^{-2}]} = [M^{-1}T^4]$$ $$\frac{[a]}{[b]} = \frac{[T^2]}{[M^{-1}T^4]} = \boxed{[MT^{-2}]}$$This is the dimension of surface tension or spring constant!
Level 3: JEE Advanced
The velocity v of water waves depends on wavelength λ, density ρ, and acceleration due to gravity g. Find the relation between them using dimensional analysis.
Solution: Let $v = k\lambda^a \rho^b g^c$ (k is dimensionless)
Dimensions:
- [v] = [LT⁻¹]
- [λ] = [L]
- [ρ] = [ML⁻³]
- [g] = [LT⁻²]
Comparing powers:
- For M: 0 = b → b = 0
- For T: -1 = -2c → c = 1/2
- For L: 1 = a - 3b + c = a + 1/2 → a = 1/2
(The actual formula is v = √(gλ/2π), where k = 1/√(2π))
Quick Reference
| Dimensional Formula | Physical Quantities |
|---|---|
| [L] | Length, wavelength, amplitude |
| [T] | Time, time period, half-life |
| [LT⁻¹] | Velocity, wave velocity |
| [LT⁻²] | Acceleration, gravity |
| [MLT⁻¹] | Momentum, impulse |
| [MLT⁻²] | Force, weight, thrust |
| [ML²T⁻²] | Work, energy, torque |
| [ML²T⁻³] | Power |
| [ML⁻¹T⁻²] | Pressure, stress |
| [ML⁻³] | Density |
| [MT⁻²] | Surface tension, spring constant |
| [ML⁻¹T⁻¹] | Viscosity |
| [T⁻¹] | Frequency, angular velocity, decay constant |
Related Topics
Within Units and Measurements
- SI Units — Foundation of dimensional analysis
- Dimensional Analysis — Applications and problem-solving
- Errors — Connecting dimensions to error propagation
- Significant Figures — Precision in dimensional calculations
Connected Chapters
- Kinematics — Dimensions in motion
- Newton’s Laws — Force dimensions
- Gravitation — Gravitational constant dimensions
- Electrostatics — Electrical quantity dimensions