Dimensional Formulae: Class 11 Physics Notes, Application & Formula

Updated on Jun 12, 2025 09:16 IST

By Jaya Sharma, Assistant Manager - Content

A dimensional formula represents a physical quantity as a product of base dimensions raised to appropriate powers. As per the NCERT and the Class 11 Notes on Units and Measurement, there are seven base dimensions:

Length: [L]
Mass: [M]
Time: [T]
Electric current: [A]
Thermodynamic temperature: [K]
Amount of substance: [mol]
Luminous intensity: [cd]

For example, the dimensional formula for velocity is $[v] = \frac{L}{T} = L^{1} T^{- 1}$ , indicating no mass, one unit of length, and negative one unit of time. Dimensional formulae are foundational for JEE Main, enabling dimensional analysis and unit consistency checks.

As per NCERT:

“The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity. For example, the dimensional formula of the volume is [M° L3 T°], and that of speed or velocity is $M^{°} L T^{- 1}$ . Similarly, $M^{°} L T^{- 2}$ is the dimensional formula of acceleration and $M L^{- 3} T^{°}$ that of mass density”

Students must practice the NCERT solutions of the Units and Measurement chapter to excel in the CBSE board examination.

Table of content

Dimensional Formular SI Base Quantities and Units*
Derivation of Dimensional Formula
Applications of Dimensional Formulae

Dimensional Formular SI Base Quantities and Units*

The following table shows some basic dimensional formulae for students to keep in mind for exams such as JEE Main and NEET:

Base quantity	Name	Symbol	Definition
Length	metre	m	The metre, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299792458 when expressed in the unit m·s⁻¹, where the second is defined in terms of the caesium frequency Δν_Cs.
Mass	kilogram	kg	The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015 × 10⁻³⁴ when expressed in the unit J·s, which is equal to kg·m²·s⁻¹, where the metre and the second are defined in terms of c and Δν_Cs.
Time	second	s	The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency Δν_Cs, the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be 9192631770 when expressed in the unit Hz, which is equal to s⁻¹.
Electric Current	ampere	A	The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge e to be 1.602176634 × 10⁻¹⁹ when expressed in the unit C, which is equal to A·s, where the second is defined in terms of Δν_Cs.
Thermodynamic Temperature	kelvin	K	The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is defined by taking the fixed numerical value of the Boltzmann constant k to be 1.380649 × 10⁻²³ when expressed in the unit J·K⁻¹, which is equal to kg·m²·s⁻²·K⁻¹, where the kilogram, metre and second are defined in terms of h, c and Δν_Cs.
Amount of Substance	mole	mol	The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.02214076 × 10²³ elementary entities. This number is the fixed numerical value of the Avogadro constant N_A, when expressed in the unit mol⁻¹ and is called the Avogadro number. The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles.
Luminous Intensity	candela	cd	The candela, symbol cd, is the SI unit of luminous intensity in a given direction. It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency 540 × 10¹² Hz, K_cd, to be 683 when expressed in the unit lm·W⁻¹, which is equal to cd·sr⁻¹·m⁻²·s³, where the kilogram, metre and second are defined in terms of h, c and Δν_Cs.

Derivation of Dimensional Formula

1. Mechanics

Velocity $v = \frac{distance}{time}$ : $[v] = [L] [T -^{1}]$
Acceleration $a = \frac{v}{t}$ : $[a] = [L] [^{T] - 2}$
Force $F = m a$ : $[F] = [M] [L] [^{T] - 2}$
Work $W = F d$ : $[W] = [M] [^{L] 2} [T -^{2}]$
Power $P = \frac{W}{t}$ : $[P] = [M] [^{L] 2} [^{T] - 3}$

2. Electromagnetism

Electric Charge $Q = I t$ : $[Q] = [A] [T]$
Potential Difference $V = \frac{W}{Q}$ : $[V] = \frac{[M] [^{L] 2} [^{T] - 2}}{[A] [T]} = [M] [^{L] 2} [^{T] - 3} [A] -^{1}$
Resistance $R = \frac{V}{I}$ : $[R] = \frac{[M] [^{L] 2} [^{T] - 3} [A] -^{1}}{[A]} = [M] [^{L] 2} [^{T] - 3} [^{A] - 2}$

3. Thermodynamics

Pressure $P = \frac{F}{A}$ : $[P] = [M] [^{L] - 1} [^{T] - 2}$
Gas Constant $R in P V = n R T$ : $[R] = [M] [^{L] 2} [^{T] - 2} [Θ] -^{1} mol -^{1}$

These derivations are important for JEE Main, where questions often require computing dimensional formulae for complex quantities.

Applications of Dimensional Formulae

Dimensional analysis is employed in physics primarily to check dimensional homogeneity in equations, to derive scaling relations via dimensionless groupings, and to perform consistent unit conversions across measurement systems.

1. Checking Dimensional Consistency

Dimensional homogeneity requires that every term in a valid physical equation share the same base-unit dimensions.

For example, in the kinetic energy formula $E = \frac{1}{2} m v^{2}$ each term has dimensions $[M] [^{L] 2} [^{T] - 2}$

2. Deriving Physical Relations

The Buckingham π theorem allows one to predict how quantities scale by forming dimensionless products without detailed theory.

For a simple pendulum, assume $T \propto L^{x} \cdot g^{y}$ . Equating $T^{1} = L^{x} {L \cdot T^{- 2}}^{y}$ yields x=½ and y=−½, hence $T = k \cdot \sqrt{\frac{L}{g}}$