Dimensional Formulae

Every physical quantity that we measure or calculate can be expressed in terms of a small set of fundamental quantities called base dimensions. In physics, the seven SI base dimensions are mass (M), length (L), time (T), electric current (I), thermodynamic temperature (Θ), amount of substance (N), and luminous intensity (J). A dimensional formula is a symbolic expression that shows exactly how a derived quantity is built from these base dimensions raised to appropriate powers. For example, the dimensional formula of velocity is written as [v]=M0L1T−1[v] = \text{M}^0\text{L}^1\text{T}^{-1}, telling us that velocity depends on one factor of length divided by one factor of time, and is independent of mass. Understanding dimensional formulae is far more than bookkeeping. They allow physicists to check whether an equation is physically consistent—a process called dimensional analysis—to convert quantities between unit systems, and even to derive unknown relationships between physical quantities from first principles. A correctly written equation must be dimensionally homogeneous: every term on both sides must share the same dimensional formula. If a formula predicts that you are adding a length to a time, it is instantly flagged as wrong, regardless of numerical values. In this article we build up the idea of dimensional formulae systematically: starting from the base dimensions, learning how to derive the formula for any quantity from its defining equation, and practising with important examples from mechanics, thermodynamics, and electromagnetism. Mastering this skill is one of the most powerful habits you can develop in physics, because it gives you a rapid, reliable way to spot errors and to reason about unfamiliar situations before doing any detailed calculation.

Key takeaways
  • Know the Base Dimensions
  • Start from the Defining Equation
  • Build Up Compound Quantities Step by Step
  • Electromagnetic Quantities
  • Checking Dimensional Homogeneity

Know the Base Dimensions

The foundation of all dimensional formulae is the set of base dimensions. In mechanics we mainly use mass M\text{M}, length L\text{L}, and time T\text{T}. Electromagnetic quantities also need electric current I\text{I}. Every derived quantity is a product of powers of these symbols: MaLbTcId…\text{M}^a\text{L}^b\text{T}^c\text{I}^d\ldots where a,b,c,d,…a,b,c,d,\ldots are rational numbers (often integers). A dimensionless quantity has all exponents equal to zero.

Start from the Defining Equation

The safest route to a dimensional formula is to use the definition or a well-known formula for the quantity. For instance, force is defined by Newton's second law: F=maF = ma. Since [m]=M[m] = \text{M} and [a]=L T−2[a] = \text{L}\,\text{T}^{-2}, we multiply: [F]=M⋅L T−2=MLT−2[F] = \text{M}\cdot\text{L}\,\text{T}^{-2} = \text{M}\text{L}\text{T}^{-2}. The rule is simple: multiplication of quantities means addition of exponents; division means subtraction of exponents; raising to a power means multiplying all exponents by that power.

Build Up Compound Quantities Step by Step

Complex quantities are handled by chaining simpler results. Energy (work) is W=FdW = Fd, so [W]=[F][d]=MLT−2⋅L=ML2T−2[W] = [F][d] = \text{M}\text{L}\text{T}^{-2}\cdot\text{L} = \text{M}\text{L}^2\text{T}^{-2}. Power is P=W/tP = W/t, giving [P]=ML2T−3[P] = \text{M}\text{L}^2\text{T}^{-3}. Pressure is p=F/Ap = F/A, so [p]=MLT−2/L2=ML−1T−2[p] = \text{M}\text{L}\text{T}^{-2}/\text{L}^2 = \text{M}\text{L}^{-1}\text{T}^{-2}. Always write intermediate results clearly so exponents are not confused.

Electromagnetic Quantities

When electric current I\text{I} enters, use Ohm's law and definitions carefully. Electric charge Q=ItQ = It, so [Q]=IT[Q] = \text{I}\text{T}. Voltage (potential difference) V=W/QV = W/Q, giving [V]=ML2T−3I−1[V] = \text{M}\text{L}^2\text{T}^{-3}\text{I}^{-1}. Resistance R=V/IR = V/I, so [R]=ML2T−3I−2[R] = \text{M}\text{L}^2\text{T}^{-3}\text{I}^{-2}. Capacitance C=Q/VC = Q/V, giving [C]=M−1L−2T4I2[C] = \text{M}^{-1}\text{L}^{-2}\text{T}^{4}\text{I}^{2}. These results look complicated, but they follow mechanically from the definitions.

Checking Dimensional Homogeneity

Once you know the dimensional formulae of all terms in an equation, verify that every term has the same formula. Consider the kinematic equation s=ut+12at2s = ut + \tfrac{1}{2}at^2. Check: [s]=L[s] = \text{L}; [ut]=LT−1⋅T=L[ut] = \text{L}\text{T}^{-1}\cdot\text{T} = \text{L}; [at2]=LT−2⋅T2=L[at^2] = \text{L}\text{T}^{-2}\cdot\text{T}^2 = \text{L}. All three terms are L\text{L}, confirming the equation is dimensionally homogeneous. If any term differed, the equation would be physically invalid.

The Key Equation

[Q]=Ma Lb Tc Id Θe Nf Jg[Q] = \text{M}^a\,\text{L}^b\,\text{T}^c\,\text{I}^d\,\Theta^e\,\text{N}^f\,\text{J}^g
  • \text{M} — Mass dimension (kilogram (kg)): Exponent aa indicates how many factors of mass appear. Negative values mean mass appears in the denominator.
  • \text{L} — Length dimension (metre (m)): Exponent bb counts factors of length; b=2b=2 gives area, b=3b=3 gives volume.
  • \text{T} — Time dimension (second (s)): Exponent cc is often negative for rates and frequencies.
  • \text{I} — Electric current dimension (ampere (A)): Enters when electromagnetic quantities such as charge, voltage, or resistance are involved.
  • \Theta — Temperature dimension (kelvin (K)): Appears in thermodynamic quantities such as entropy (ML2T−2Θ−1\text{M}\text{L}^2\text{T}^{-2}\Theta^{-1}).

Worked Example

Problem. Derive the dimensional formula for the gravitational constant GG from Newton's law of universal gravitation: F = \frac{Gm_1 m_2}{r^2}

[G]=M−1L3T−2[G] = \text{M}^{-1}\text{L}^3\text{T}^{-2}

Answer. The dimensional formula of the gravitational constant is M−1L3T−2\text{M}^{-1}\text{L}^3\text{T}^{-2}.

Figure
Diagram
Dimensional Formula Tree for EnergyEnergy W[M L² T⁻²]Force F[M L T⁻²]Distance d[L]Mass m[M]Acceleration a[L T⁻²]Length L[L]Time² T²[T²]W = F·d = (ma)·d → [M][L T⁻²][L] = [M L² T⁻²]
Origins · 1 minJoseph Fourier and how this was discovered
JF
Joseph Fourier
1768–1830 · French

Joseph Fourier is credited as the first person to systematically use dimensional reasoning in physics. In his landmark 1822 work 'Théorie analytique de la chaleur' (The Analytical Theory of Heat), he insisted that every term in a physical equation must have the same dimensions—a principle he called 'dimensional homogeneity'. He explicitly introduced the idea that physical quantities carry dimensional exponents and that consistent physics requires these to balance. His framework is the direct ancestor of modern dimensional analysis and dimensional formulae. Fourier participated in Napoleon's Egyptian campaign in 1798 as a scientific adviser and later served as a prefect (regional governor) while continuing his mathematical research—writing some of his greatest physics while managing local administration.

  • Applications of Dimensional Analysis — Dimensional formulae are the building blocks used in dimensional analysis to derive equations, check consistency, and convert units.
  • SI Base Units and Fundamental Quantities — Dimensional formulae are expressed in terms of SI base dimensions (M, L, T, I, Θ, N, J), so understanding the base units is a prerequisite.
  • Significant Figures and Measurement Error — Both topics sit within Units, Dimensions & Measurement Techniques and are applied together when reporting and checking physical results.
  • Newton's Laws of Motion — Newton's second law F=maF=ma is the most common starting point for deriving dimensional formulae of mechanical quantities like force, energy, and pressure.
  • Buckingham π Theorem — This advanced theorem formalises dimensional analysis by showing how dimensional formulae can be combined into dimensionless groups to find physical relationships.