What "dimension" means in physics
NCERT's definition in §1.4 is the cleanest one to memorise: the dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity. Square brackets around a quantity — like $[F]$ — signal that we are talking about its dimensions, not its numerical magnitude.
The seven base dimensions correspond one-for-one with the seven SI base quantities established in the SI system of units:
| Base quantity | Dimension symbol | SI base unit |
|---|---|---|
| Length | [L] | metre (m) |
| Mass | [M] | kilogram (kg) |
| Time | [T] | second (s) |
| Electric current | [A] | ampere (A) |
| Thermodynamic temperature | [K] | kelvin (K) |
| Amount of substance | [mol] | mole (mol) |
| Luminous intensity | [cd] | candela (cd) |
In mechanics, only $[M]$, $[L]$ and $[T]$ are needed. Volume, the product of three lengths, has dimensions $[L^3]$ — but NCERT explicitly writes its full dimensional signature as $[M^0 L^3 T^0]$, the zero exponents stating that mass and time do not enter. Force is mass times acceleration:
$$[F] = [M] \cdot \frac{[L]}{[T]^2} = [M L T^{-2}]$$NCERT closes §1.4 with a critical remark: magnitudes are not considered when writing dimensions; it is the quality of the type of physical quantity that enters. Initial velocity, final velocity, average velocity and speed all share the dimensions $[L T^{-1}]$. A dimension does not distinguish a velocity from a speed, or an instantaneous value from an average.
"The nature of a physical quantity is described by its dimensions."
NCERT Class 11 Physics, §1.4
Dimensional formula vs dimensional equation
NCERT §1.5 separates two terms that students often blur together.
A dimensional formula is the expression that shows which base quantities, and to what powers, build a given derived quantity. Examples lifted directly from NCERT §1.5:
- Volume: $[M^0 L^3 T^0]$
- Speed or velocity: $[M^0 L T^{-1}]$
- Acceleration: $[M^0 L T^{-2}]$
- Mass density: $[M L^{-3} T^0]$
A dimensional equation is what you get when you set the symbol of the physical quantity (in square brackets) equal to its dimensional formula. NCERT writes them as:
$$[V] = [M^0 L^3 T^0], \quad [v] = [M^0 L T^{-1}], \quad [F] = [M L T^{-2}], \quad [\rho] = [M L^{-3} T^0]$$The compact way to state the distinction: a formula is the right-hand side; an equation is the formula bracketed with the symbol on the left. NEET questions usually ask "what is the dimensional formula of X?" — they want the right-hand side, but the answer is conventionally written as a dimensional equation.
Four more worked formulas
Work along with each derivation. The skill is learning to read the defining equation as a chain of multiplications and divisions of base quantities.
Find the dimensional formula of linear momentum $p = mv$.
$[p] = [m] \cdot [v] = [M] \cdot [L T^{-1}] = [M L T^{-1}]$. SI unit: kg m s$^{-1}$.
Find the dimensional formula of work $W = F \cdot d$.
$[W] = [F] \cdot [d] = [M L T^{-2}] \cdot [L] = [M L^2 T^{-2}]$. SI unit: kg m$^2$ s$^{-2}$, named joule (J). Kinetic energy $\tfrac{1}{2}mv^2$ gives $[M] \cdot [L T^{-1}]^2 = [M L^2 T^{-2}]$ — the same formula, as it must, because work-energy theorem demands it. NCERT Example 1.4 ruled out three "kinetic energy formulas" using exactly this dimensional check.
Find the dimensional formula of pressure $P = F/A$.
$[P] = [F] / [A] = [M L T^{-2}] / [L^2] = [M L^{-1} T^{-2}]$. SI unit: kg m$^{-1}$ s$^{-2}$, named pascal (Pa).
Find the dimensional formula of power $P = W/t$.
$[P] = [W] / [t] = [M L^2 T^{-2}] / [T] = [M L^2 T^{-3}]$. SI unit: kg m$^2$ s$^{-3}$, named watt (W).
Master table — 35 NEET-essential quantities
The list below collects every dimensional formula NEET has tested, or is likely to test, across mechanics, thermal physics, electromagnetism and modern physics. Use it as a reference and as a self-test: cover the right-hand columns, read the defining relation, and reconstruct the formula before peeking.
| Quantity | Defining relation | Dimensional formula | SI unit |
|---|---|---|---|
| Mechanics — kinematics & dynamics | |||
| Velocity / speed | $v = \dfrac{\text{length}}{\text{time}}$ | $[M^0 L T^{-1}]$ | m s$^{-1}$ |
| Acceleration | $a = \dfrac{v}{t}$ | $[M^0 L T^{-2}]$ | m s$^{-2}$ |
| Force | $F = ma$ | $[M L T^{-2}]$ | N (kg m s$^{-2}$) |
| Linear momentum | $p = mv$ | $[M L T^{-1}]$ | kg m s$^{-1}$ |
| Impulse | $J = F \cdot t$ | $[M L T^{-1}]$ | N s |
| Work / energy / heat | $W = F \cdot d$ | $[M L^2 T^{-2}]$ | J |
| Power | $P = W/t$ | $[M L^2 T^{-3}]$ | W |
| Pressure / stress | $P = F/A$ | $[M L^{-1} T^{-2}]$ | Pa (N m$^{-2}$) |
| Young's modulus / bulk modulus | stress / strain | $[M L^{-1} T^{-2}]$ | Pa |
| Density | $\rho = m/V$ | $[M L^{-3} T^0]$ | kg m$^{-3}$ |
| Frequency | $\nu = 1/T$ | $[M^0 L^0 T^{-1}]$ | Hz (s$^{-1}$) |
| Angular velocity / angular frequency | $\omega = \theta/t$ | $[M^0 L^0 T^{-1}]$ | rad s$^{-1}$ |
| Angular momentum | $L = I\omega = mvr$ | $[M L^2 T^{-1}]$ | kg m$^2$ s$^{-1}$ |
| Torque / moment of force | $\tau = r \times F$ | $[M L^2 T^{-2}]$ | N m |
| Moment of inertia | $I = mr^2$ | $[M L^2 T^0]$ | kg m$^2$ |
| Surface tension | $\gamma = F/L$ | $[M L^0 T^{-2}]$ | N m$^{-1}$ |
| Coefficient of viscosity | $F = \eta A \,dv/dx$ | $[M L^{-1} T^{-1}]$ | Pa s |
| Universal gravitational constant $G$ | $F = G m_1 m_2 / r^2$ | $[M^{-1} L^3 T^{-2}]$ | N m$^2$ kg$^{-2}$ |
| Gravitational potential | $V_g = U/m$ | $[M^0 L^2 T^{-2}]$ | J kg$^{-1}$ |
| Spring constant | $F = kx$ | $[M L^0 T^{-2}]$ | N m$^{-1}$ |
| Thermal physics | |||
| Temperature | — | $[K]$ | K |
| Specific heat capacity | $c = Q/(m \Delta T)$ | $[M^0 L^2 T^{-2} K^{-1}]$ | J kg$^{-1}$ K$^{-1}$ |
| Latent heat | $L = Q/m$ | $[M^0 L^2 T^{-2}]$ | J kg$^{-1}$ |
| Boltzmann constant $k_B$ | $E = k_B T$ | $[M L^2 T^{-2} K^{-1}]$ | J K$^{-1}$ |
| Universal gas constant $R$ | $PV = nRT$ | $[M L^2 T^{-2} K^{-1} \text{mol}^{-1}]$ | J mol$^{-1}$ K$^{-1}$ |
| Coefficient of thermal conductivity | $Q/t = -kA \,dT/dx$ | $[M L T^{-3} K^{-1}]$ | W m$^{-1}$ K$^{-1}$ |
| Stefan–Boltzmann constant $\sigma$ | $E = \sigma T^4$ | $[M L^0 T^{-3} K^{-4}]$ | W m$^{-2}$ K$^{-4}$ |
| Electricity & magnetism | |||
| Electric current | — | $[A]$ | A |
| Electric charge | $q = It$ | $[M^0 L^0 T A]$ | C (A s) |
| Electric potential / EMF | $V = W/q$ | $[M L^2 T^{-3} A^{-1}]$ | V (J C$^{-1}$) |
| Electric field | $E = F/q$ | $[M L T^{-3} A^{-1}]$ | V m$^{-1}$ |
| Electrical resistance | $R = V/I$ | $[M L^2 T^{-3} A^{-2}]$ | $\Omega$ (V A$^{-1}$) |
| Capacitance | $C = q/V$ | $[M^{-1} L^{-2} T^4 A^2]$ | F (C V$^{-1}$) |
| Permittivity of free space $\varepsilon_0$ | $F = q_1 q_2 / (4\pi\varepsilon_0 r^2)$ | $[M^{-1} L^{-3} T^4 A^2]$ | C$^2$ N$^{-1}$ m$^{-2}$ |
| Permeability of free space $\mu_0$ | $F/L = \mu_0 I_1 I_2 / (2\pi r)$ | $[M L T^{-2} A^{-2}]$ | T m A$^{-1}$ (or H m$^{-1}$) |
| Magnetic field $B$ | $F = qvB$ | $[M L^0 T^{-2} A^{-1}]$ | T (tesla) |
| Magnetic flux $\Phi_B$ | $\Phi_B = B \cdot A$ | $[M L^2 T^{-2} A^{-1}]$ | Wb (V s) |
| Self / mutual inductance | $\varepsilon = -L \,dI/dt$ | $[M L^2 T^{-2} A^{-2}]$ | H (henry) |
| Modern physics & optics | |||
| Planck constant $h$ | $E = h\nu$ | $[M L^2 T^{-1}]$ | J s |
| Refractive index $n$ | $n = c/v$ | $[M^0 L^0 T^0]$ | dimensionless |
| Wavelength | — | $[M^0 L T^0]$ | m |
| Wave number $\bar{\nu}$ | $\bar{\nu} = 1/\lambda$ | $[M^0 L^{-1} T^0]$ | m$^{-1}$ |
Three observations to encode now. Every "rate" carries one extra $T^{-1}$ over its parent — angular velocity is angle per time, power is energy per time, current is charge per time. Every "per area" or "per volume" inherits negative powers of $L$ — pressure $L^{-1}$, density $L^{-3}$. Electromagnetic quantities are the ones involving $[A]$ — permeability, capacitance, flux, inductance.
Dimensionless quantities — units without dimensions
A quantity is dimensionless if every base exponent in its dimensional formula is zero — written $[M^0 L^0 T^0]$. NCERT §1.6.1 explains why these appear: they are always ratios of two quantities of the same kind, so the units cancel. Memorise the canonical list.
| Dimensionless quantity | Definition / ratio | Unit (if any) |
|---|---|---|
| Plane angle $\theta$ | arc length / radius | radian (rad) |
| Solid angle $\Omega$ | area / radius$^2$ | steradian (sr) |
| Strain | change in length / original length | — |
| Refractive index $n$ | speed of light in vacuum / in medium | — |
| Relative density (specific gravity) | density of substance / density of water | — |
| Dielectric constant $\kappa$ (relative permittivity $\varepsilon_r$) | $\varepsilon / \varepsilon_0$ | — |
| Relative permeability $\mu_r$ | $\mu / \mu_0$ | — |
| Coefficient of friction $\mu$ | friction force / normal force | — |
| Mechanical advantage / efficiency | ratio of forces or energies | — |
| Poisson's ratio $\sigma$ | lateral strain / longitudinal strain | — |
| Fine-structure constant $\alpha$ | $e^2/(4\pi\varepsilon_0 \hbar c) \approx 1/137$ | — |
| Arguments of $\sin$, $\cos$, $\log$, $e^x$ | must be pure numbers | — |
Same dimensions, different physical meaning
Dimensional formulae are coarse — they record the type of base quantities involved, not the structure built from them. So several physically distinct quantities can share an identical formula. NEET examiners love testing this; NCERT explicitly warns in §1.6.2 that "It does not distinguish between the physical quantities having same dimensions." The four most-tested clashes are:
| Common dimensional formula | Quantities that share it | Why they differ physically |
|---|---|---|
| $[M L^2 T^{-2}]$ | Work, kinetic energy, potential energy, heat, torque | Energy is a scalar describing transfer; torque is a vector cross-product $\mathbf{r} \times \mathbf{F}$. Joule is the unit of energy; torque uses N m (never J). |
| $[M L^{-1} T^{-2}]$ | Pressure, stress, Young's / bulk / shear modulus, energy density | Stress is force per area; modulus is stress over (dimensionless) strain — same dimensions. Energy density is energy per volume, which also collapses to this formula. |
| $[M L^2 T^{-1}]$ | Angular momentum, Planck constant | $L = I\omega$ and $h$ via $E = h\nu$. The shared dimensions are why Bohr's quantisation rule $L = n\hbar$ works — but angular momentum is a vector, $h$ a scalar constant. |
| $[M^0 L^0 T^{-1}]$ | Frequency, angular frequency, decay constant, velocity gradient | $\nu = 1/T$ (Hz); $\omega = 2\pi\nu$ (rad s$^{-1}$); decay constant $\lambda$ in $e^{-\lambda t}$ (s$^{-1}$). Same $[T^{-1}]$, different physical context. |
How to find the dimensional formula of an unfamiliar quantity
You will not always be given a memorised formula. The reliable procedure: start from a defining equation, isolate the unknown, and read off dimensions from the rest. Four worked examples below cover the four kinds of derivation NEET tests.
From $F = \gamma L$, find $[\gamma]$.
Surface tension $\gamma$ is force per unit length along the boundary of a liquid surface. Rearranging, $\gamma = F/L$. Substituting dimensions:
$$[\gamma] = \frac{[F]}{[L]} = \frac{[M L T^{-2}]}{[L]} = [M L^0 T^{-2}] = [M T^{-2}]$$
SI unit: N m$^{-1}$. The $L^0$ is conventional — you can write either $[M T^{-2}]$ or $[M L^0 T^{-2}]$, the latter signals deliberately that length cancelled.
From Newton's law $F = G m_1 m_2 / r^2$, find $[G]$ (NIOS Terminal Q.4).
$G = F r^2 / (m_1 m_2)$, so:
$$[G] = \frac{[M L T^{-2}] \cdot [L^2]}{[M]^2} = [M^{-1} L^3 T^{-2}]$$
SI unit: N m$^2$ kg$^{-2}$.
From $E = h\nu$, find $[h]$.
$h = E/\nu$, so $[h] = [M L^2 T^{-2}] / [T^{-1}] = [M L^2 T^{-1}]$. SI unit: J s — same dimensional formula as angular momentum.
From $F = \eta A \,(dv/dx)$, find $[\eta]$.
Velocity gradient $dv/dx$ has dimensions $[L T^{-1}] / [L] = [T^{-1}]$. So:
$$[\eta] = \frac{[M L T^{-2}]}{[L^2] \cdot [T^{-1}]} = [M L^{-1} T^{-1}]$$
SI unit: Pa s. The CGS unit is the poise = g cm$^{-1}$ s$^{-1}$.
The same procedure handles every NEET-bank quantity: write the defining equation, isolate the unknown, substitute dimensions for every symbol on the right-hand side. The full set of dimensional-analysis applications — checking equations for homogeneity, deriving the form of an unknown relation, and converting between unit systems — is treated in the sibling article.
What this subtopic locked in
- Seven base dimensions. $[M], [L], [T], [A], [K], [\text{mol}], [\text{cd}]$. Mechanics needs only the first three.
- Formula vs equation. Dimensional formula is the right-hand side; dimensional equation sets it equal to $[X]$.
- 35-row master table. Mechanics, thermal, electromagnetism and modern physics — every dimensional formula NEET expects, grouped for revision.
- Dimensionless quantities. Plane angle, solid angle, strain, refractive index, relative density, dielectric constant, $\mu_r$, friction coefficient, Poisson's ratio — and arguments of $\sin$, $\log$, $\exp$.
- Same dimensions, different physics. Work/torque, pressure/stress/Young's modulus, angular momentum/Planck constant, frequency/angular frequency/decay constant.
- Derivation procedure. Start from the defining equation, isolate the unknown, substitute dimensions for every symbol on the right.