Physics
Units and Measurements
Dimensional Formulas

Units and Measurements · JEE & NEET Physics

Dimensional Formulas: notes and previous year questions

What dimensions are, how to find them, which quantities share them, and the principle of homogeneity.

84 JEE Main questions (2003–2026)
11 JEE Advanced questions (1998–2026)
13 NEET questions (2004–2024)

Dimensional Formulas in short

Dimensions are the powers of the base quantities (M, L, T, I, θ, N, J) in a quantity.
Square brackets mean “the dimensions of”; powers of 0 and 1 are usually not written.
A negative power means division: speed is [LT⁻¹], length divided by time.
A unit is the standard you measure with; the dimension is the kind of quantity. Changing the unit never changes the dimension.

1What are dimensions?

Every physical quantity is built from a few base quantities: mass, length, time and so on. The dimensions of a quantity are the powers to which the base quantities are raised to make it.

Think of a recipe. A cake recipe says how much flour, sugar and eggs go in. A dimensional formula says how much mass, length and time go into a quantity. Speed is length divided by time, so it has length to the power 1 and time to the power −1, and no mass at all:

[v] = [\mathrm{M}^0\,\mathrm{L}^1\,\mathrm{T}^{-1}] = [\mathrm{L}\mathrm{T}^{-1}]

square brackets mean “the dimensions of”

A negative power means the base quantity divides; a power of 0 means it is not used. Powers of 0 and 1 are usually not written, so $[\mathrm{M}^0\mathrm{L}^1\mathrm{T}^{-1}]$ is simply $[\mathrm{LT}^{-1}]$ .

There are seven base quantities, so there are seven dimension symbols:

Base quantity	Dimension	SI unit
Mass	$[\mathrm{M}]$	kilogram (kg)
Length	$[\mathrm{L}]$	metre (m)
Time	$[\mathrm{T}]$	second (s)
Electric current	$[\mathrm{I}]$ (some books write $[\mathrm{A}]$ )	ampere (A)
Temperature	$[\theta]$ (some books write $[\mathrm{K}]$ )	kelvin (K)
Amount of substance	$[\mathrm{N}]$ (or $[\text{mol}]$ )	mole (mol)
Luminous intensity	$[\mathrm{J}]$ (or $[\text{cd}]$ )	candela (cd)

2Unit or dimension?

A unit is the standard you measure with: metre, kilometre, centimetre. A dimension says what kind of quantity something is: a length, a speed, a force. The two are easy to mix up, but they answer different questions.

Quantity	SI unit	Dimensional formula
Velocity	m/s	$[\mathrm{M}^0\mathrm{L}^1\mathrm{T}^{-1}] = [\mathrm{LT}^{-1}]$
Force	newton (N)	$[\mathrm{MLT}^{-2}]$
Energy	joule (J)	$[\mathrm{ML}^2\mathrm{T}^{-2}]$

3Method 1: from the formula

Start from the formula that defines the quantity, and put in the dimensions of each part. Three power rules do all the work:

Multiply two quantities → add their powers.
Divide → subtract the powers.
Raise to a power → multiply the powers.
Pure numbers such as $\tfrac12$ , 2 and $\pi$ have no dimensions, so they drop out.

Using the same rules you get every quantity in mechanics:

Quantity	Defined as	Dimensional formula	SI unit
Area	$l^2$	$[\mathrm{L}^2]$	m²
Volume	$l^3$	$[\mathrm{L}^3]$	m³
Density	$m/V$	$[\mathrm{ML}^{-3}]$	kg/m³
Velocity	$s/t$	$[\mathrm{LT}^{-1}]$	m/s
Acceleration	$v/t$	$[\mathrm{LT}^{-2}]$	m/s²
Momentum	$mv$	$[\mathrm{MLT}^{-1}]$	kg m/s
Force	$ma$	$[\mathrm{MLT}^{-2}]$	N
Impulse	$Ft$	$[\mathrm{MLT}^{-1}]$	N s
Work, energy	$Fs$	$[\mathrm{ML}^2\mathrm{T}^{-2}]$	J
Power	$W/t$	$[\mathrm{ML}^2\mathrm{T}^{-3}]$	W
Pressure	$F/A$	$[\mathrm{ML}^{-1}\mathrm{T}^{-2}]$	Pa
Angular velocity	$\theta/t$	$[\mathrm{T}^{-1}]$	rad/s
Angular acceleration	$\omega/t$	$[\mathrm{T}^{-2}]$	rad/s²
Moment of inertia	$mr^2$	$[\mathrm{ML}^2]$	kg m²
Torque	$rF$	$[\mathrm{ML}^2\mathrm{T}^{-2}]$	N m
Angular momentum	$I\omega$	$[\mathrm{ML}^2\mathrm{T}^{-1}]$	kg m²/s

4Method 2: from the SI unit

Write the SI unit in base units, then swap each base unit for its letter: kg → M, m → L, s → T (and A → I, K → θ, mol → N).

Heat. Temperature brings in $\theta$ :

Quantity	Dimensional formula	SI unit
Temperature	$[\theta]$	K
Heat $Q$	$[\mathrm{ML}^2\mathrm{T}^{-2}]$	J
Specific heat $c$	$[\mathrm{L}^2\mathrm{T}^{-2}\theta^{-1}]$	J/(kg K)
Latent heat $L$	$[\mathrm{L}^2\mathrm{T}^{-2}]$	J/kg
Thermal conductivity $k$	$[\mathrm{MLT}^{-3}\theta^{-1}]$	W/(m K)
Stefan's constant $\sigma$	$[\mathrm{MT}^{-3}\theta^{-4}]$	W/(m² K⁴)
Boltzmann constant $k_B$	$[\mathrm{ML}^2\mathrm{T}^{-2}\theta^{-1}]$	J/K
Gas constant $R$	$[\mathrm{ML}^2\mathrm{T}^{-2}\theta^{-1}\mathrm{N}^{-1}]$	J/(mol K)

Electricity and magnetism. Current brings in $\mathrm{I}$ . Charge is current × time, so $[q] = [\mathrm{IT}]$ ; the rest follow from their units:

Quantity	Dimensional formula	SI unit
Current $I$	$[\mathrm{I}]$	A
Charge $q$	$[\mathrm{IT}]$	C = A s
Voltage, emf $V$	$[\mathrm{ML}^2\mathrm{T}^{-3}\mathrm{I}^{-1}]$	V = J/C
Resistance $R$	$[\mathrm{ML}^2\mathrm{T}^{-3}\mathrm{I}^{-2}]$	Ω = V/A
Capacitance $C$	$[\mathrm{M}^{-1}\mathrm{L}^{-2}\mathrm{T}^4\mathrm{I}^2]$	F = C/V
Electric field $E$	$[\mathrm{MLT}^{-3}\mathrm{I}^{-1}]$	N/C = V/m
Magnetic field $B$	$[\mathrm{MT}^{-2}\mathrm{I}^{-1}]$	T
Magnetic flux $\Phi$	$[\mathrm{ML}^2\mathrm{T}^{-2}\mathrm{I}^{-1}]$	Wb
Inductance $L$	$[\mathrm{ML}^2\mathrm{T}^{-2}\mathrm{I}^{-2}]$	H = Wb/A
Permittivity $\varepsilon_0$	$[\mathrm{M}^{-1}\mathrm{L}^{-3}\mathrm{T}^4\mathrm{I}^2]$	F/m
Permeability $\mu_0$	$[\mathrm{MLT}^{-2}\mathrm{I}^{-2}]$	H/m

Waves, modern physics and constants:

Quantity	Dimensional formula	SI unit
Frequency $f$	$[\mathrm{T}^{-1}]$	Hz
Wavelength $\lambda$ , amplitude $A$	$[\mathrm{L}]$	m
Wave speed $v$	$[\mathrm{LT}^{-1}]$	m/s
Time period, half-life	$[\mathrm{T}]$	s
Spring constant $k$	$[\mathrm{MT}^{-2}]$	N/m
Surface tension	$[\mathrm{MT}^{-2}]$	N/m
Viscosity $\eta$	$[\mathrm{ML}^{-1}\mathrm{T}^{-1}]$	Pa s
Planck's constant $h$	$[\mathrm{ML}^2\mathrm{T}^{-1}]$	J s
Work function $\phi$	$[\mathrm{ML}^2\mathrm{T}^{-2}]$	J or eV
Decay constant $\lambda$	$[\mathrm{T}^{-1}]$	s⁻¹
Gravitational constant $G$	$[\mathrm{M}^{-1}\mathrm{L}^3\mathrm{T}^{-2}]$	N m²/kg²
Acceleration due to gravity $g$	$[\mathrm{LT}^{-2}]$	m/s²
Avogadro's number $N_A$	$[\mathrm{N}^{-1}]$	mol⁻¹

5Quantities with the same dimensions

Two different quantities can have exactly the same dimensional formula. Work ( $F \times s$ ), kinetic energy ( $\tfrac12 mv^2$ ), heat and torque ( $r \times F$ ) are all $[\mathrm{ML}^2\mathrm{T}^{-2}]$ .

Families worth learning together:

Dimensional formula	Quantities
$[\mathrm{L}]$	length, wavelength, amplitude
$[\mathrm{T}]$	time, time period, half-life
$[\mathrm{LT}^{-1}]$	velocity, wave speed
$[\mathrm{LT}^{-2}]$	acceleration, acceleration due to gravity
$[\mathrm{MLT}^{-1}]$	momentum, impulse
$[\mathrm{MLT}^{-2}]$	force, weight, thrust
$[\mathrm{ML}^2\mathrm{T}^{-2}]$	work, energy, heat, torque
$[\mathrm{ML}^2\mathrm{T}^{-3}]$	power
$[\mathrm{ML}^2\mathrm{T}^{-1}]$	angular momentum, Planck's constant
$[\mathrm{ML}^{-1}\mathrm{T}^{-2}]$	pressure, stress
$[\mathrm{ML}^{-3}]$	density
$[\mathrm{MT}^{-2}]$	surface tension, spring constant
$[\mathrm{ML}^{-1}\mathrm{T}^{-1}]$	viscosity
$[\mathrm{T}^{-1}]$	frequency, angular velocity, decay constant

6Dimensionless quantities

Some quantities have no dimensions: every power is zero. We write them as $[\mathrm{M}^0\mathrm{L}^0\mathrm{T}^0]$ , or simply $[1]$ .

Quantity	Why it is dimensionless
Pure numbers	1, 2, $\pi$ , $e$ are just numbers
Strain	$\Delta L / L$ : length ÷ length
Angle, and $\sin\theta$ , $\cos\theta$ , $\tan\theta$	arc ÷ radius; ratios of two sides
Relative density	density ÷ density of water
Refractive index	$n = c/v$ : speed ÷ speed
Coefficient of friction	$\mu = f/N$ : force ÷ force
Poisson's ratio	lateral strain ÷ longitudinal strain
Reynolds number	$Re = \rho v L / \eta$ : all the dimensions cancel

7The principle of homogeneity

Principle of homogeneity: in a correct physical equation, every term on both sides has the same dimensions. You can only add or subtract like with like: metres to metres, never metres to seconds.

Pure numbers such as $\tfrac12$ , 2 and $\pi$ do not affect the check.
What goes inside $\sin$ , $\cos$ , $\log$ or $e^x$ must be dimensionless. In $y = A\sin(\omega t)$ , $\omega t$ has no dimensions, so $[\omega] = [\mathrm{T}^{-1}]$ .

Summary

Key ideas

Dimensions are the powers of the base quantities (M, L, T, I, θ, N, J) in a quantity.
Square brackets mean “the dimensions of”; powers of 0 and 1 are usually not written.
A negative power means division: speed is [LT⁻¹], length divided by time.
A unit is the standard you measure with; the dimension is the kind of quantity. Changing the unit never changes the dimension.
Method 1: put the dimensions into the defining formula. Multiplying adds powers, dividing subtracts them, raising to a power multiplies them.
Method 2: write the SI unit in base units, then swap kg → M, m → L, s → T.
Per area multiplies by [L⁻²]; per length multiplies by [L⁻¹].
Different quantities can share a formula: work, energy, heat and torque are all [ML²T⁻²], but torque is written in N m, never J.
Pure numbers, ratios of like quantities and angles are dimensionless: [M⁰L⁰T⁰].
An angle has a unit (the radian) but no dimensions.
Principle of homogeneity: every term in a correct equation has the same dimensions.
Whatever sits inside sin, cos, log or eˣ must be dimensionless. A dimensionally correct equation can still be wrong by a pure number.

Every equation

Velocity: $[v] = [\mathrm{M}^0\mathrm{L}^1\mathrm{T}^{-1}] = [\mathrm{LT}^{-1}]$
Acceleration: $[a] = [\mathrm{LT}^{-2}]$
Force: $[F] = [\mathrm{MLT}^{-2}]$
Momentum, impulse: $[p] = [F t] = [\mathrm{MLT}^{-1}]$
Work, energy, heat, torque: $[\mathrm{ML}^2\mathrm{T}^{-2}]$
Kinetic energy: $[\tfrac12 mv^2] = [\mathrm{M}][\mathrm{LT}^{-1}]^2 = [\mathrm{ML}^2\mathrm{T}^{-2}]$
Power: $[P] = [F][v] = [\mathrm{ML}^2\mathrm{T}^{-3}]$
Pressure, stress: $[P] = [\mathrm{ML}^{-1}\mathrm{T}^{-2}]$
Density: $[\rho] = [\mathrm{ML}^{-3}]$
Surface tension, spring constant: $[\mathrm{MT}^{-2}]$
Viscosity: $[\eta] = [\mathrm{ML}^{-1}\mathrm{T}^{-1}]$
Frequency, angular velocity: $[\mathrm{T}^{-1}]$
Moment of inertia: $[I] = [\mathrm{ML}^2]$
Angular momentum, Planck's constant: $[\mathrm{ML}^2\mathrm{T}^{-1}]$
Gravitational constant: $[G] = [\mathrm{M}^{-1}\mathrm{L}^3\mathrm{T}^{-2}]$
Charge: $[q] = [\mathrm{IT}]$
Voltage: $[V] = [\mathrm{ML}^2\mathrm{T}^{-3}\mathrm{I}^{-1}]$
Resistance: $[R] = [\mathrm{ML}^2\mathrm{T}^{-3}\mathrm{I}^{-2}]$
Capacitance: $[C] = [\mathrm{M}^{-1}\mathrm{L}^{-2}\mathrm{T}^4\mathrm{I}^2]$
Electric field: $[E] = [\mathrm{MLT}^{-3}\mathrm{I}^{-1}]$
Magnetic field: $[B] = [\mathrm{MT}^{-2}\mathrm{I}^{-1}]$
Magnetic flux: $[\Phi] = [\mathrm{ML}^2\mathrm{T}^{-2}\mathrm{I}^{-1}]$
Inductance: $[L] = [\mathrm{ML}^2\mathrm{T}^{-2}\mathrm{I}^{-2}]$
Permittivity: $[\varepsilon_0] = [\mathrm{M}^{-1}\mathrm{L}^{-3}\mathrm{T}^4\mathrm{I}^2]$
Permeability: $[\mu_0] = [\mathrm{MLT}^{-2}\mathrm{I}^{-2}]$
Specific heat: $[c] = [\mathrm{L}^2\mathrm{T}^{-2}\theta^{-1}]$
Latent heat: $[L] = [\mathrm{L}^2\mathrm{T}^{-2}]$
Thermal conductivity: $[k] = [\mathrm{MLT}^{-3}\theta^{-1}]$
Stefan's constant: $[\sigma] = [\mathrm{MT}^{-3}\theta^{-4}]$
Boltzmann constant: $[k_B] = [\mathrm{ML}^2\mathrm{T}^{-2}\theta^{-1}]$
Gas constant: $[R] = [\mathrm{ML}^2\mathrm{T}^{-2}\theta^{-1}\mathrm{N}^{-1}]$
Avogadro's number: $[N_A] = [\mathrm{N}^{-1}]$
Dimensionless: $[\text{angle}] = [\text{strain}] = [\mathrm{M}^0\mathrm{L}^0\mathrm{T}^0] = [1]$
Homogeneity check: $[s] = [ut] = [\tfrac12 at^2] = [\mathrm{L}]$

Previous year questions with solutions

Real JEE and NEET questions on dimensional formulas. Try each one before you open the solution.

Q1JEE Main 2026One correct option

The potential energy of a particle changes with distance $x$ from a fixed origin as $V=\frac{A\sqrt{x}}{x+B}$ , where $A$ and $B$ are constant with appropriate dimensions. The dimensions of $AB$ are $\_\_\_\_$

A $[M^{1}L^{5/2}T^{-2}]$
B $[M^{3/2}L^{5/2}T^{-2}]$
C $[M^{1}L^{2}T^{-2}]$
D $[M^{1}L^{7/2}T^{-2}]$

Show answer and solution

Answer: Option D

Take the addition first: $B$ is added to $x$ , so $[B] = [\mathrm{L}]$ and the whole bracket is $[\mathrm{L}]$ .

$V$ here is a potential energy, $[\mathrm{ML^{2}T^{-2}}]$ , so $[A] = \dfrac{[V][\mathrm{L}]}{[\sqrt{x}]} = \dfrac{[\mathrm{ML^{2}T^{-2}}][\mathrm{L}]}{[\mathrm{L^{1/2}}]} = [\mathrm{ML^{5/2}T^{-2}}]$ .

Multiplying by $[B] = [\mathrm{L}]$ raises the power of length by one more: $[AB] = [\mathrm{ML^{7/2}T^{-2}}]$ , which is option D.

Q2JEE Advanced 2026Numerical answer

In a new system of units, the units of mass, length, time and current are $5 \mathrm{kg},5 m,5 s$ and 5 A , respectively. If ${\mu}_{0}$ and ${\epsilon}_{0}$ are the permeability and permittivity of free space, respectively, then in this new system of units, the magnitude of one SI unit of $\sqrt{{\mu}_{0}/{\epsilon}_{0}}$ , is :

Show answer and solution

Answer: 25

First find the dimensional formula. $\dfrac{[\mu_{0}]}{[\epsilon_{0}]} = \dfrac{[\mathrm{MLT^{-2}A^{-2}}]}{[\mathrm{M^{-1}L^{-3}T^{4}A^{2}}]} = [\mathrm{M^{2}L^{4}T^{-6}A^{-4}}]$ , and the square root of that is $[\mathrm{ML^{2}T^{-3}A^{-2}}]$ — a resistance.

Now change systems. A quantity's number goes up when the unit gets smaller, so $n_{2} = n_{1}\left(\dfrac{M_{1}}{M_{2}}\right)^{1}\left(\dfrac{L_{1}}{L_{2}}\right)^{2}\left(\dfrac{T_{1}}{T_{2}}\right)^{-3}\left(\dfrac{A_{1}}{A_{2}}\right)^{-2}$ , with every ratio equal to $\tfrac{1}{5}$ .

The powers add up to $1 + 2 - 3 - 2 = -2$ , so $n_{2} = \left(\tfrac{1}{5}\right)^{-2} = 25$ .

Q3NEET 2022One correct option

Plane angle and solid angle have

AUnits but no dimensions
BDimensions but no units
CNo units and no dimensions
DBoth units and dimensions

Show answer and solution

Answer: Option A

A plane angle is an arc divided by a radius, $\dfrac{[\mathrm{L}]}{[\mathrm{L}]}$ , and a solid angle is an area divided by the square of a radius, $\dfrac{[\mathrm{L^{2}}]}{[\mathrm{L^{2}}]}$ . Both are ratios of like quantities, so both come out as $[\mathrm{M^{0}L^{0}T^{0}}]$ .

They are still given names to write down, the radian and the steradian, so they have units. That is option A.

Practice questions, easy to hard

Three questions from the dimensional formulas practice ladder: one easy, one medium, one hard.

Q4One correct option

Heat also creeps along a solid bar. The rate at which it flows is $\dfrac{Q}{t} = kA\dfrac{\Delta T}{l}$ , where $l$ is the length of the bar and $k$ is the thermal conductivity.

Rearrange: $k = \dfrac{(Q/t)\,l}{A\,\Delta T}$ . The rate $Q/t$ is a power.

What is the dimensional formula of $k$ ?

A $[\mathrm{ML^{2}T^{-3}K^{-1}}]$
B $[\mathrm{MT^{-3}K^{-1}}]$
C $[\mathrm{MLT^{-3}K^{-1}}]$
D $[\mathrm{ML^{-1}T^{-3}K^{-1}}]$

Show answer and solution

Answer: Option C

Start with the power $[\mathrm{ML^{2}T^{-3}}]$ , multiply by the length $[\mathrm{L}]$ to reach $[\mathrm{ML^{3}T^{-3}}]$ , then divide by the area $[\mathrm{L^{2}}]$ to come back down to $[\mathrm{MLT^{-3}}]$ . The temperature difference underneath adds $[\mathrm{K^{-1}}]$ , giving $[\mathrm{MLT^{-3}K^{-1}}]$ .

Q5One correct option

Back to electricity for two quantities the exam questions below need. An electric field is the force felt per unit charge, $E = \dfrac{F}{q}$ .

With $[F] = [\mathrm{MLT^{-2}}]$ and $[q] = [\mathrm{AT}]$ , what is the dimensional formula of $E$ ?

A $[\mathrm{MLT^{-2}A^{-1}}]$
B $[\mathrm{ML^{2}T^{-3}A^{-1}}]$
C $[\mathrm{MLT^{-3}A^{-1}}]$
D $[\mathrm{MT^{-3}A^{-1}}]$

Show answer and solution

Answer: Option C

Dividing by $[\mathrm{AT}]$ sends the ampere underneath and drops the power of time from $-2$ to $-3$ : $[E] = [\mathrm{MLT^{-3}A^{-1}}]$ . Its unit can be written $\mathrm{N\,C^{-1}}$ or $\mathrm{V\,m^{-1}}$ , and both translate to the same thing.