Dimensional formulae and SI units play a crucial role in understanding the nature of physical quantities and their relationships. By expressing quantities in terms of fundamental dimensions such as mass, length, and time, dimensional analysis helps in identifying whether different physical quantities have the same dimensions. Quantities having the same dimensions can often be compared, related, or substituted in equations, making this concept essential for verifying formulas, solving numerical problems, and strengthening problem-solving skills in Class 11 Physics and Competitive Exams.
- Dimensions of Physical Quantities
- Dimensional Formulae and Dimensional Equations
- Dimensions of Fundamental Quantities
- Understanding Dimensions with Example
- Dimensional Equations and Dimensional Formula
- Important Dimensions of Physics Quantities
- Dimensional Formula for Different Physical Quantities and their SI Units
- I. Dimensional Formulae of Physical Quantities from Mechanics & Properties of Matter
- II. Dimensional Formulae of Physical Quantities from Heat & Thermodynamics
- III. Dimensional Formulae of Physical Quantities from Electricity & Magnetism
- IV. Dimensional Formulae of Physical Quantities from Atomic & Nuclear Physics
- Physical Quantities Having the Same Dimensional Formula
- Different Types of Variables and Constants in Physics
- Uses of Dimensional Equations
- FAQs (Frequently Asked Questions) Very Short Questions and Answers
Dimensions of Physical Quantities
The derived units of all the physical quantities can be suitably expressed in terms of the fundamental units of mass, length and time and time raised to some power. For example, if units of mass, length and time are denoted by bracketed capital letters [M], [L] and [T], then for area (= length × breadth), we have
$$
\text{area} = [L] \times [L] = [L^{2}]
$$
Further, as in expressing area, units of mass and time do not occur, we write
$$
\text{area} = [M^{0} L^{2} T^{0}]
$$
The dimensions of area are zero in mass, two in length and zero in time.
Similarly, for velocity, we have
$$
\text{velocity} = \frac{\text{distance}}{\text{time}} = \frac{[L]}{[T]} = [L T^{-1}]
$$
$$
= [M^{0} L T^{-1}]
$$
The dimensions of velocity are zero in mass, one in length and minus one in time.
Hence, the dimensions of a physical quantity are the powers to which the fundamental units of mass, length and time have to be raised in order to obtain its units.
Dimensional Formulae and Dimensional Equations
The expression $[M^{0} L^{2} T^{0}]$ for area and $[M^{0} L T^{-1}]$ for velocity are said to be the dimensional formulae of area and velocity respectively.
The dimensional formula of a physical quantity provides two important informations. For example, the dimensional formula $[M^{0} L T^{-1}]$ for velocity tells that
(i) the unit of velocity depends upon the unit of length and time and is independent of the unit of mass.
(ii) in the unit of velocity, the powers of L and T i.e. the units of length and time are 1 and –1 respectively.
Hence, dimensional formula of a physical quantity may be defined as the expression that indicates which of the fundamental units of mass, length and time enter into the derived unit of that quantity and with what powers.
If we represent velocity by $[v]$, then an equation such as
$$
[v] = [M^{0} L T^{-1}]
$$
is known as dimensional equation.
Thus, the equation obtained, when a physical quantity is equated with its dimensional formula, is known as dimensional equation.
In general, in the dimensional equation
$$
[X] = [M^{a} L^{b} T^{c}]
$$
the right hand side represents the dimensional formula of physical quantity X, whose dimensions in mass, length and time are a, b and c respectively.
Dimensions of Fundamental Quantities
The dimensions of a physical quantity are the powers to which the unit of fundamental quantities are raised to represent that quantity.
The seven fundamental quantities and their respective dimensions are:
| S.No. | Physical Quantity | Dimension |
|---|---|---|
| 1 | Length | [L] |
| 2 | Mass | [M] |
| 3 | Time | [T] |
| 4 | Temperature | [K] |
| 5 | Electric Current | [A] |
| 6 | Luminous Intensity | [Cd] |
| 7 | Amount of Substance | [Mol] |
** Note: Two supplementary fundamental quantities that is plane angle and solid angle have no dimensions.
Understanding Dimensions with Example
To make the concept clearer, consider the physical quantity force:
Force = Mass × Acceleration
Since Velocity is given by:
Velocity = Length/Time
and acceleration is given by:
Acceleration=Velocity/Time=Length/Time2
Force = Mass x Acceleration
Therefore, Force=Mass×Length/Time2
Force=Mass×Length×(Time)−2
Thus, the dimensions of force are:
Force=[MLT−2]
Dimensional Equations and Dimensional Formula
The physical quantity that is expressed in terms of base quantities is enclosed in square brackets to indicate that the equation represents dimensions and not magnitudes.
Dimensional Equation : The equation obtained by equating a physical quantity with its dimensions formula is called dimensional equation of the given physical quantity
For Example : The dimensional equation of Force is :
[Force]=[MLT−2]
Dimensional Formula : If we consider only the right-hand side (RHS) of the equation, it is called the dimensional formula:
Dimensional Formula of Force=[MLT−2]
Important Dimensions of Physics Quantities
Here are some commonly used physical quantities with their dimensional formulas:
| Physical Quantity | Dimensional Formula |
|---|---|
| Velocity | [LT−1] |
| Acceleration | [LT−2] |
| Momentum | Mas x Velocity = [MLT−1] |
| Work or Energy | Force x Distance = [ML2T−2] |
| Power | Work/Time = [ML2T−3] |
| Pressure | Force/Area = [ML-1T−2] |
Dimensional Formula for Different Physical Quantities and their SI Units
The dimensional formula of a physical quantity can be obtained by defining its relation with other physical quantities and then expressing these physical quantities in terms of fundamental units of mass [M], length [L] and time [T].
In the following table, the dimensional formulae of a few physical quantities have been deduced after defining their relations with other physical quantities. The SI units of these quantities have also been given.
I. Dimensional Formulae of Physical Quantities from Mechanics & Properties of Matter
| Physical quantity | Relation with other physical quantities | Dimensional formula | SI unit |
|---|---|---|---|
| Area | length × breadth | $L \times L = L^{2} = [M^{0} L^{2} T^{0}]$ | m² |
| Volume | length × breadth × height | $L \times L \times L = L^{3} = [M^{0} L^{3} T^{0}]$ | m³ |
| Density | mass/volume | $M/L^{3} = [M L^{-3} T^{0}]$ | kg m⁻³ |
| Specific gravity | density of a material/density of water at 4° C | Dimensionless | — |
| Specific volume | volume/mass | $L^{3}/M = [M^{-1} L^{3} T^{0}]$ | m³ kg⁻¹ |
| Linear speed or velocity | distance/time | $L/T = [M^{0} L T^{-1}]$ | m s⁻¹ |
| Linear acceleration | velocity/time | $L/T^{-1} = [M^{0} L T^{-2}]$ | m s⁻² |
| Linear momentum | mass × velocity | $M \times L T^{-1} = [M L T^{-1}]$ | kg m s⁻¹ |
| Force | mass × acceleration | $M \times L T^{-2} = [M L T^{-2}]$ | N (newton) |
| Force constant | force/increase in length | $[M L T^{-2}]/L = [M^{0} L^{0} T^{-2}]$ | N m⁻¹ |
| Pressure | force/area | $M L T^{-2}/L^{2} = [M L^{-1} T^{-2}]$ | N m⁻² or Pa (pascal) |
| Thrust | force | $[M L T^{-2}]$ | N |
| Tension | force | $[M L T^{-2}]$ | N |
| Work | force × distance | $M L T^{-2} \times L = [M L^{2} T^{-2}]$ | J (joule) |
| Energy (mechanical, heat, light, etc) | work | $[M L^{2} T^{-2}]$ | J |
| Power | work/time | $M L^{2} T^{-2}/T = [M L^{2} T^{-3}]$ | W (watt) |
| Pressure gradient | pressure/distance | $M L^{-1} T^{-2}/L = [M L^{-2} T^{-2}]$ | N m⁻³ |
| Moment of force | force × L distance | $M L T^{-2} \times L = [M L^{2} T^{-2}]$ | N m |
| Gravitational constant | force × (distance)²/mass × mass | $M L T^{-2} \times L^{2}/M \times M = [M^{-1} L^{3} T^{-2}]$ | N m² kg⁻² |
| Hubble constant | velocity of recession/distance | $L T^{-1}/L = [M^{0} L^{0} T^{-1}]$ | s⁻¹ |
| Impulse | force × time | $M L T^{-2} \times T = [M L T^{-1}]$ | N s |
| Stress | force/area | $M L T^{-2}/L^{2} = [M L^{-1} T^{-2}]$ | N m⁻² |
| Strain | change in dimension/original dimension | Dimensionless | — |
| Coefficient of elasticity | stress/strain | $M L^{-1} T^{-2}/1 = [M L^{-1} T^{-2}]$ | N m⁻² |
| Surface Tension | force/length | $M L T^{-2}/L = [M L^{0} T^{-2}]$ | N m⁻¹ |
| Surface energy | energy/area | $M L^{2} T^{-2}/L^{2} = [M L^{0} T^{-2}]$ | J m⁻² |
II. Dimensional Formulae of Physical Quantities from Heat & Thermodynamics
| Physical quantity | Relation with other physical quantities | Dimensional formula | SI unit |
|---|---|---|---|
| Temperature | Fundamental unit | $[M^{0} L^{0} T^{0} K]$ | K (kelvin) |
| Specific heat capacity | quantity of heat / (mass × temperature) | $[M L^{2} T^{-2}] / [M][K]$ | J kg⁻¹ K⁻¹ |
| Latent heat | quantity of heat / mass | $[M L^{2} T^{-2}] / [M]$ | J kg⁻¹ |
| Universal gas constant | (pressure × volume) / temperature | $[M L^{-1} T^{-2}][L^{3}] / [K]$ | J K⁻¹ |
| Avogadro number | number of molecules per mole | Dimensionless | mol⁻¹ |
| Boltzmann’s constant | universal gas constant / Avogadro number | $[M L^{2} T^{-2} K^{-1}]$ | J K⁻¹ |
| Coefficient of thermal expansion | change in dimension / (dimension × temperature) | $[L] / [L][K]$ | K⁻¹ |
| Temperature gradient | temperature / distance | $[K] / [L]$ | K m⁻¹ |
| Coefficient of thermal conductivity | quantity of heat / (area × time × temp. gradient) | $[M L^{2} T^{-2}] / [L^{2}][T][L^{-1} K]$ | W m⁻¹ K⁻¹ |
| Wien’s constant | wavelength × temperature | $[L][K] = [M^{0} L T^{0} K]$ | m K |
| Stefan’s constant | energy emitted / (area × time × temperature⁴) | $[M L^{2} T^{-2}] / [L^{2}][T][K^{4}]$ | W m⁻² K⁻⁴ |
| Solar constant | energy emitted by the sun / (area × time) | $[M L^{2} T^{-2}] / [L^{2}][T]$ | W m⁻² |
III. Dimensional Formulae of Physical Quantities from Electricity & Magnetism
| Physical quantity | Relation with other physical quantities | Dimensional formula | SI unit |
|---|---|---|---|
| Electric current (I) | Fundamental unit | $[M^{0} L^{0} T^{0} A]$ | A (ampere) |
| Electric current density (j) | current/area | $[A] / [L^{2}]$ | A m⁻² |
| Electric charge (q) | current × time | $A \times T = [M^{0} L^{0} T A]$ | C (coulomb) |
| Electric potential (V) | work/charge | $[M L^{2} T^{-2}] / [A T]$ | V (volt) |
| Electric field intensity (E) | force/charge | $[M L T^{-2}] / [A T]$ | N C⁻¹ |
| Permittivity of free space (ε₀) | charge × charge / (force × distance²) | $[A T]^{2} / [M L T^{-2} L^{2}]$ | C² N m⁻² |
| Electric flux (φ_E) | electric field × area | $[M L T^{-3} A^{-1} L^{2}]$ | N m² C⁻¹ |
| Electric capacitance (C) | charge/potential difference | $[A T] / [M L^{2} T^{-2} A^{-1}]$ | F (farad) |
| Surface charge density (σ) | charge/area | $[A T] / [L^{2}]$ | C m⁻² |
| Volume charge density (ρ) | charge/volume | $[A T] / [L^{3}]$ | C m⁻³ |
| Electric dipole moment (p) | charge × length | $[A T] / [L]$ | C m |
| Electric resistance (R) | potential difference/current | $[M L^{2} T^{-2} A^{-1}] / [A]$ | Ω (ohm) |
| Resistivity (ρ) | resistance × area/length | $[M L^{2} T^{-2} A^{-2}] \times L / L^{2}$ | Ω m |
| Electric conductance (G) | 1/resistance | $1 / [M L^{2} T^{-2} A^{-2}]$ | S (siemen) or Ω⁻¹ |
| Conductivity (σ) | 1/resistivity | $1 / [M L^{3} T^{-2} A^{-2}]$ | S m⁻¹ or Ω⁻¹ m⁻¹ |
| Coefficient of self induction (L) or mutual induction (M) | e.m.f. × time / current | $[M L^{2} T^{-2} A^{-1}] / [A]$ | H (henry) |
| Inductive reactance (X_L) | ωL | $[T^{-1}] \times [M L^{2} T^{-2} A^{-2}]$ | Ω |
| Capacitive reactance (X_C) | 1/(ωC) | $1 / ([T^{-1}] [M^{-1} L^{-2} T^{4} A^{2}])$ | Ω |
| Power factor (cos φ) | Trigonometric ratio | Dimensionless | No unit |
| Resonant angular frequency (ω₀) | 1/√(LC) | $[M^{0} L^{0} T^{-1}]$ | rad s⁻¹ or Hz |
| Quality factor (Q) | ω₀L / R | Dimensionless | No unit |
| Permeability of free space (μ₀) | 2π × force × distance / (current² × length) | $[M L T^{-2}][L] / [A^{2}][L]$ | N A⁻² or Wb A⁻¹ m⁻¹ |
| Magnetic pole strength (m) | √(4π × force × distance² / μ₀) | $([M L T^{-2}][L]^{2})^{1/2} / [M L T^{-2} A^{-2}]$ | A m |
| Magnetic dipole moment (p_m) | pole strength × distance | $[M^{0} L T^{0} A][L]$ | A m² |
| Magnetic induction (B) | μ₀ × current / (2π × distance) | $[M L T^{-2} A^{-2}][A] / [L]$ | N m⁻¹ A⁻¹ or tesla (T) |
| Magnetic flux (Φ_B) | B × area | $[M L^{0} T^{-2} A^{-1}][L^{2}]$ | N m A⁻¹ or weber (Wb) |
| Coefficient of self inductance (L) or mutual inductance (M) | magnetic flux / current | $[M L^{2} T^{-2} A^{-1}] / [A]$ | H (henry) |
| Magnetic intensity (H) | magnetic induction / μ₀ | $[M L^{0} T^{-2} A^{-1}] / [M L T^{-2} A^{-2}]$ | A m⁻¹ |
| Intensity of magnetisation (I) | magnetic moment / volume | $[M^{0} L^{2} T^{0} A] / [L^{3}]$ | A m⁻¹ |
| Coercivity | H (opposing) | $[M^{0} L^{-1} T^{0} A]$ | A m⁻¹ |
| Retentivity | I (residual) | $[M^{0} L^{-1} T^{0} A]$ | A m⁻¹ |
IV. Dimensional Formulae of Physical Quantities from Atomic & Nuclear Physics
| Physical quantity | Relation with other physical quantities | Dimensional formula | SI unit |
|---|---|---|---|
| Refractive index | speed of light in vacuum / speed of light in medium | Dimensionless | — |
| Work function | energy | $[M L^{2} T^{-2}]$ | J |
| Planck’s constant | energy / frequency | $[M L^{2} T^{-2}] / [T^{-1}]$ | J s |
| Wavelength | length of a wave | $[M^{0} L T^{0}]$ | m |
| Wave number | 1 / wavelength | $1 / [L]$ | m⁻¹ |
| Rydberg’s constant | mass × charge⁴ / (8 ε₀² × h³ × velocity) | Complex expression (see book) | m⁻¹ |
| Faraday constant | Avogadro number × e | $[A T]$ | C |
| Packing fraction | mass defect / atomic number | $[M]$ | kg nucleon⁻¹ |
| Decay constant | 0.693 / half-life | $1 / [T]$ | s⁻¹ |
Physical Quantities Having the Same Dimensional Formula
| Dimension | Quantities |
|---|---|
| M⁰L⁰T⁻¹ | Frequency, angular frequency, angular velocity, velocity gradient, decay constant |
| M¹L²T⁻² | Work, internal energy, potential energy, kinetic energy, torque, moment of force |
| M¹L⁻¹T⁻² | Pressure, stress, Young’s modulus, bulk modulus, modulus of rigidity, energy density |
| M¹L¹T⁻¹ | Momentum, impulse |
| M⁰L¹T⁻² | Acceleration due to gravity, gravitational field intensity |
| M¹L¹T⁻² | Thrust, force, weight, energy gradient |
| M¹L²T⁻¹ | Angular momentum, Planck’s constant |
| M¹L⁰T⁻² | Surface tension, surface energy (energy per unit area) |
| M⁰L⁰T⁰ | Strain, refractive index, relative density, angle, solid angle, distance gradient, relative permittivity, relative permeability |
| M⁰L²T⁻² | Latent heat, gravitational potential |
| M⁰L²T⁻²θ⁻¹ | Thermal capacity, gas constant, Boltzmann constant, entropy |
| M⁰L⁰T¹ | L/R, √LC, RC (L = Inductance, R = Resistance, C = Capacitance) |
| M¹L²T⁻² | I2Rt, VIt, V2/Rt,qV, 1/2LI2, 1/2q2/C, 1/2CV2, Various electrical quantities involving L=inductance, C=capacitance, q=charge, R=resistance, I=Current and V=voltage |
Different Types of Variables and Constants in Physics
From the study of dimensional formulae of physical quantities, we can divide them into four categories:
(i) Dimensional variables. The quantities like area, volume, velocity, force, etc possess dimensions and do not have a constant value. Such quantities are called dimensional variables.
(ii) Non-dimensional variables. The quantities like angle, specific gravity, strain, etc neither possess dimensions nor they have a constant value. Such quantities are called non-dimensional variables.
(iii) Dimensional constants. The quantities like gravitational constant, Planck’s constant, etc possess dimensions and also have a constant value. They are called dimensional constants.
(iv) Non-dimensional constants. The constant quantities having no dimensions are called non-dimensional constants. These include pure numbers $1, 2, 3, 4 \dots$, $\pi$, $e$ ($= 2.718$) and all trigonometrical functions.
Uses of Dimensional Equations
The dimensional equations have got the following three uses:
1. To check the correctness of a physical equation.
2. To derive the relation between different physical quantities involved in a physical phenomenon.
3. To change from one system of units to another.
FAQs (Frequently Asked Questions) Very Short Questions and Answers
What Are Dimensional Constants?
The physical quantities with dimensions and a fixed value are called dimensional constants. For example, gravitational constant (G), Planck’s constant (h), universal gas constant (R), velocity of light in a vacuum (C), etc.
What Are Dimensionless Quantities?
Dimensionless quantities are those which do not have dimensions but have a fixed value.
Dimensionless quantities without units: Pure numbers, π, e, sin θ, cos θ, tan θ etc.
Dimensionless quantities with units: Angular displacement – radian, Joule’s constant – joule/calorie, etc.
What Are Dimensional Variables?
Dimensional variables are those physical quantities which have dimensions and do not have a fixed value. For example, velocity, acceleration, force, work, power, etc.
What Are the Dimensionless Variables?
Dimensionless variables are those physical quantities which do not have dimensions and do not have a fixed value. For example, specific gravity, refractive index, the coefficient of friction, Poisson’s ratio, etc.
Why are dimensions important in physics?
Dimensions help in verifying equations, deriving new relationships, and converting units.
What is the difference between the dimensional formula and the dimensional equation?
The dimensional formula consists only of the dimensions of a quantity, whereas the dimensional equation shows how the quantity is derived in terms of base quantities.
Can two different physical quantities have the same dimensions?
Yes, for example, work and energy both have the same dimensions [ML²T⁻²].
Can a physical equation be correct even if its dimensions do not match on both sides?
No, for an equation to be dimensionally correct, both sides must have the same dimensions.
Why do we use dimensions in checking the correctness of an equation?
Because dimensions remain unchanged in all systems of units, making them useful for equation verification.
What is the dimension of gravitational constant (G)?
The dimension of gravitational constant (G) is [M⁻¹L³T⁻²].
Why is dimensional analysis important in physics?
Dimensional analysis helps check the correctness of equations, derive new formulas, and convert units between different measurement systems.
Why do work and torque have the same dimensions but different physical meanings?
Though both have dimensions M¹L²T⁻², work is a scalar quantity representing energy transfer, whereas torque is a vector quantity causing rotational motion.
What are dimensionless quantities? Give examples.
Dimensionless quantities have no physical dimensions. Examples include strain, refractive index, relative density, and solid angle.
Write Physical Quantities Having the Same Dimensional Formula
1. Impulse and momentum.
2. Work, torque, the moment of force, energy.
3. Angular momentum, Planck’s constant, rotational impulse.
4. Stress, pressure, modulus of elasticity, energy density.
5. Force constant, surface tension, surface energy.
6. Angular velocity, frequency, velocity gradient.
7. Gravitational potential, latent heat.
8. Thermal capacity, entropy, universal gas constant and Boltzmann’s constant.
9. Force, thrust.
10. Power, luminous flux.
Units and Measurements – Chapter Summary
The chapter Units and Measurements provides the fundamental framework for all topics in physics by explaining how physical quantities are defined, measured, and expressed using standard units. It begins with the introduction of physical quantities, followed by their classification into fundamental quantities and derived quantities, along with a detailed study of SI units, supplementary units, and commonly used systems of units (CGS, MKS, FPS). You can explore each concept in detail through dedicated sections on units, symbols, and prefixes.
The chapter further develops measurement concepts such as accuracy and precision, and explains different types of errors including absolute error, relative error, percentage error, and propagation of errors (error in sum, difference, multiplication, division, and power). These topics are essential for understanding how uncertainties affect results and are widely used in numerical problem-solving and experimental physics.
Another important part of the chapter is significant figures and rounding off rules, which help maintain the correct level of precision in calculations. Finally, dimensional analysis and dimensional formulae are introduced to check the correctness of equations, derive relationships, identify quantities having the same dimensions, and perform unit conversions. This chapter forms the base for all numerical and theoretical concepts in physics for JEE Exam.