Dimensional Formula

About Dimensional Formula

Let us first define dimension before studying the dimensional formula. In math, a dimension is a measurement of length, width, or height extended in one direction. It is a measure of a point / line extended in one direction, according to dimension definition. Every shape in our environment has proportions. In mathematics, there’s no specific dimensional formula for the idea of dimension. The power to which the fundamental units are elevated to obtain one unit of any physical quantity is called its dimension. Let's have a look at the dimensional formula and see some examples at the conclusion. Get the list of Maths Formulas.

What Is the Dimensional Formula?

The phrase expressing the powers to which the fundamental units must be increased in order to obtain 1 unit of a derived quantity is known as the dimensional formula of any quantity. If Q is any physical quantity, the dimensional formula is represented by,

List of Dimensional Formula table

Q = M^aL^bT^c

where M, L, and T are the basic dimensions of mass, length, and time, and a, b, and c are the exponents of each.

Physical quantity	Unit	Dimensional formula
Length	m	L
Mass	kg	M
Time	s	T
Acceleration or acceleration due to gravity	ms^-2	LT^-2
Angle(arc/radius)	rad	M⁰L⁰T⁰
Angular displacement	rad	M⁰L⁰T⁰
Angular impulse(torque x time)	Nms	ML²T^-1
Angular momentum(Iω)	kgm²s^-1	ML²T^-1
Angular velocity(angle/time)	rads^-1	T^-1
Area(length x breadth)	m²	L²
Boltzmann's constant	JK^-r	M²L²T^-2θ^-1
Bulk modulus(ΔP X (V/ΔV))	Nm^-2,Pa	M¹L^-1T^-2
Calorific Value	Jkg^-1	L²T^-2
Coefficient of linear or areal or volume expansion	⁰C^-1or K^-1	θ^-1
Coefficient of surface tension(force/length)	Nm^-1or Jm^-2	MT^-2
Coefficient of thermal conductivity	Wm^-1K^-1	MLT^-3θ^-1
Coefficient of viscosity(F = η x A x (dv/dx))	poise	ML^-1T^-1
Compressibility(1/bulk modulus)	Paⁱ,m²N^-2	M^-1LT²
Density(mass/volume)	kgm^-3	ML^-3
Displacement,wavelength,focal length	m	L
Electric Capacitance(charge/potential)	CV^-1,farad	M^-1L^-2T⁴I²
Electric Conductance(1/resistance)	Ohm^-1,or mho or siemen	M^-1L^-2T³I²
Electric Conductivity(1/resistivity)	siemen/metre or Sm^-1	M^-1L^-3T³I²
Electric charge or quantity of electric charge (current × time)	coulomb	IT
Electric current	ampere	I
Electric dipole moment (charge × distance)	Cm	LTI
Electric dipole moment (charge × distance)	NC^-1,Vm^-1	MLT^-3I^-1
Electric resistance (potential difference/current)	ohm	ML²T^-3I^-2
Emf (or) electric potential (work/charge)	volt	ML²T^-3I^-1
Energy (capacity to do work)	joule	ML²T^-2
Energy density (energy/volume)	Jm^-3	ML^-1T^-2
Entropy (ΔS = ΔQ/T)	Jθ^–1	ML²T^-2θ^–1
Force (mass x acceleration)	newton (N)	MLT^-2
Force constant or spring constant (force/extension)	Nm^–1	MT^-2
Frequency (1/period)	Hz	T^-1
Gravitational potential (work/mass)	Jkg^–1	L²T^-2
Heat (energy)	J or calorie	ML²T^-2
Illumination (Illuminance)	lux (lumen/metre²)	MT^-3
Impulse (force x time)	Ns or kgms^-1)	MLT^-1
Inductance (L) (energy = LI² or Coefficient of self-induction	henry (H)	ML²T^-2I^-2
Intensity of gravitational field (F/m)	Nkg^–1	L¹T^-2
Intensity of magnetization (I)	Am^–1	L^-1I
Joule’s constant or mechanical equivalent of heat	Jcal^–1	M⁰L⁰T⁰
Latent heat (Q = mL)	Jkg^–1	M⁰L²T^-2
Linear density (mass per unit length)	Kgm^–1	ML^-1
Luminous flux	lumen or (Js^–1)	ML²T^-3
Magnetic dipole moment	Am²	L²I
Magnetic flux (magnetic induction x area)	weber (Wb)	ML²T^-2I^-1
Magnetic induction (F = Bil)	NI^–1m^–1 or T	MT^-2I^-1
Magnetic pole strength	Am (ampere–meter)	LI
Modulus of elasticity (stress/strain)	Nm^–2, Pa	ML^-1T^-2
Moment of inertia (mass × radius²)	Kgm²	ML²
Momentum (mass × velocity)	Kgms^-1	MLT^-1
Permeability of free space (μo=4πFd₂m₁m₂)(μo=4πFd₂m₁m₂)	Hm^–1 or NA^-2	MLT^-2I^-2
Permittivity of free space (εo=Q₁Q₂4πFd₂)(εo=Q₁Q₂4πFd₂)	Fm^–1 or C²N^–1m^–2	M^-1L^-3T⁴I²
Planck’s constant (energy/frequency)	Js	ML²T^-1
Poisson’s ratio (lateral strain/longitudinal strain)	-	M⁰L⁰T⁰
Power (work/time)	Js^-1 or watt(W)	ML²T^-3
Pressure (force/area)	Nm^-2 or Pa	ML^-1T^-2
Pressure coefficient or volume coefficient	^oC^–1 or θ^–1	θ^–1
Pressure head	m	M⁰LT⁰
Radioactivity	disintegrations per second	M⁰L⁰T^-1
Ratio of specific heats	-	M⁰L⁰T⁰
Refractive index	-	M⁰L⁰T⁰
Resistivity or specific resistance	Ω–m	ML^-3T^-3I^-2
Specific conductance or conductivity (1/specific resistance)	siemen/metre or Sm^–1	M^-1L^-3T³I²
Specific entropy (1/entropy)	KJ^–1	M^-1L^-2T²θ
Specific gravity (density of the substance/density of water)	-	M⁰L⁰T⁰
Specific heat (Q = mst)	Jkg^–1θ^–1	M⁰L²T^-2θ^–1
Specific volume (1/density)	m³kg^–1	M^-1L³
Speed (distance/time)	ms^-1	LT^-1
Stefan’s constant (heat energyarea×time×temperature) (heat energyarea×time×temperature4)	Wm^-2θ^–4	ML⁰T^-3θ^–4
Strain (change in dimension/original dimension)	-	M⁰L⁰T⁰
Stress (restoring force/area)	Nm^-2Pa	ML^-1T^-2
Surface energy density (energy/area)	Jm^-2	MT^-2
Temperature	⁰C or θ	M⁰L⁰T⁰θ
Temperature gradient (change in temperaturedistance)(change in temperaturedistance)	⁰Cm^-1 or θm^-1	M⁰L⁰T⁰θ
Thermal capacity (mass × specific heat)	Jθ^-1	ML²T^-2θ^-1
Time period	second	T
Torque or moment of force (force × distance)	Nm	ML²T^-2
Universal gas constant (work/temperature)	Jmol^-1θ^-1	ML²T^-2θ^-1
Universal gravitational constant (F=G.m₁m₂d₂)(F=G.m₁m₂d₂)	Nm₂kg_-2	M_-1L³T^-2
Velocity (displacement/time)	ms_-1	LT_-1
Velocity gradient (dv/dx)	s_-1	T^-1
Volume (length × breadth × height)	m₃	L³
Water equivalent	kg	ML⁰T⁰
Work (force × displacement)	J	ML²T^-2
Decay constant	s^-1	M⁰L⁰T^-1
Potential energy	J	M¹L²T^-2
Kinetic energy	J	M¹L²T^-2

Dimensional Formula and Dimensional Equations

In terms of dimensions, a dimensional equation is an equation that connects fundamental and derived units. The three base dimensions in mechanics are length, mass, time, temperature, and electric current, with the fundamental units being metre, kilogramme, second, ampere, kelvin, mole, and candela. In any dimensional equation, the dimensional formula of separate values is employed to construct a link between them. The following is an example of a dimensional equation:

Dimensional formula (equation) for the area:

Area = length × breadth

= length × length

= [L] × [L]

= [L]²

⇒ Dimensional formula (equation) for area (A) = [L² M⁰ T⁰]

Applications of Dimensional Formula

It is used to verify the correctness of an equation. It is used to derive the relationship between different physical quantities. It is used to convert from one system of units to another for any given quantity.