What Are Dimensions? A Beginner's Guide

The concept of dimension in mathematics and physics refers to the independent physical nature assigned to each foundational quantity used to describe physical phenomena, such as length, mass, and time. These dimensions serve as the basis for formulating the units and relationships of all measurable physical quantities.

Mathematical Structure of Dimensions and Units in Physical Quantities

Every physical quantity is characterized by the manner in which it depends on a set of fundamental physical quantities known as base quantities. This dependence is represented by expressing the quantity as a monomial product of the powers of base quantities. If a physical quantity $Q$ depends on base quantities $A, B, C, \ldots$ as $Q = k A^x B^y C^z \ldots$, the exponents $(x, y, z, \ldots)$ are termed the dimensions of $Q$ with respect to those base quantities.

In mechanics, the International System of Units (SI) prescribes length $(L)$, mass $(M)$, and time $(T)$ as the independent base quantities for all classical physical measurements. Thus, any mechanical quantity $Q$ can be expressed with the dimensional formula: $Q = [M^a L^b T^c]$, where $a$, $b$, and $c$ are real numbers (often integers) indicating the powers of mass, length, and time, respectively, required to describe $Q$.

Classification and Expression of Units: Fundamental and Derived

A base or fundamental unit is the internationally accepted standard unit for a fundamental physical quantity. For example, the SI unit of mass is the kilogram $(\mathrm{kg})$, length is the metre $(\mathrm{m})$, and time is the second $(\mathrm{s})$. All other units are termed derived units, and they are constructed algebraically by combining the base units according to the defining relations of the derived quantity.

A unit is a fixed quantity used as a standard for measurement of a dimensioned physical quantity. For instance, the unit of area is $\mathrm{m}^2$, derived from the product of two length units: $\mathrm{m} \times \mathrm{m}$.

Dimensional Formula and Dimensional Equation of Physical Quantities

The dimensional formula of a physical quantity is an expression that shows it as a product of powers of the base quantities (in SI: $M$, $L$, $T$, etc.). For example, consider area $A$, defined as length $\times$ breadth: $A = l \times b$.

The dimensions of length and breadth are each $[L^1]$. Multiplying, $A = [L^1][L^1] = [L^2]$. Expressed in the standard dimensional form including all base quantities, $A = [M^0L^2T^0]$.

The dimensional equation is the statement of equality between the physical quantity and its dimensional formula. For area, $\text{Area} = [M^0 L^2 T^0]$.

Examples of Dimensional Analysis: Detailed Stepwise Expansion

Example: Derive the dimensional formula of velocity.

Velocity ($v$) = Displacement / Time

Let the dimension of displacement be $[L^1]$ and time be $[T^1]$.

Express velocity as: $v = \dfrac{\text{Displacement}}{\text{Time}} = \dfrac{[L^1]}{[T^1]}$.

Apply the law of exponents: $[L^1] \times [T^{-1}] = [L^1 T^{-1}]$.

Including mass, which does not appear in the definition, $v = [M^0 L^1 T^{-1}]$ as the dimensional formula of velocity.

Example: Find the dimensional formula of force.

Force ($F$) is given by Newton’s second law as $F = m \times a$, where $m$ is mass and $a$ is acceleration.

First, write the dimensional formula of mass: $[M^1]$.

Now, acceleration is velocity per unit time, so: $a = \dfrac{\text{Velocity}}{\text{Time}} = \dfrac{[L^1 T^{-1}]}{[T^1]} = [L^1 T^{-2}]$.

Multiply to obtain the formula for force: $[M^1] \times [L^1 T^{-2}] = [M^1 L^1 T^{-2}]$.

Thus, the dimensional formula for force in the SI system is $[M^1 L^1 T^{-2}]$.

Dimensional Formulae for Standard Quantities in Classical Mechanics

The following are standard dimensional formulae of common quantities encountered in examinations:

Work, Energy: $[M^1 L^2 T^{-2}]$
Power: $[M^1 L^2 T^{-3}]$
Momentum: $[M^1 L^1 T^{-1}]$
Pressure: $[M^1 L^{-1} T^{-2}]$
Density: $[M^1 L^{-3} T^0]$
Surface Tension: $[M^1 L^0 T^{-2}]$
Frequency: $[M^0 L^0 T^{-1}]$
Volume: $[M^0 L^3 T^0]$

Special Units: Plane Angle and Solid Angle

There exist supplementary SI units not expressible in terms of $M$, $L$, $T$: the radian (plane angle) and the steradian (solid angle).

For a plane angle $\theta$, $\theta = \dfrac{l}{r}$, where $l$ is the length of the arc and $r$ is its radius. Both $l$ and $r$ have dimensions $[L^1]$, so their ratio is dimensionless; thus, the radian has the dimensional formula $[M^0 L^0 T^0]$.

For a solid angle $\Omega$, $\Omega = \dfrac{\Delta S}{r^2}$, with surface area $\Delta S\ [L^2]$ and $r^2\ [L^2]$, making the steradian dimensionless as well.

Systems of Units: SI, CGS, MKS, and FPS

The SI system structures all base and derived quantities systematically using the metre, kilogram, and second. Alternative systems such as the CGS (centimetre, gram, second) and FPS (foot, pound, second) have historic and regional importance, but for examination and modern scientific communication, SI is universally employed. Only one standard unit exists for each quantity in SI, enabling worldwide consistency in scientific notation and unit conversions.

Distinction Between Dimension and Unit

Dimension refers to the independent physical character (such as mass, length, or time) involved in defining a quantity, without reference to any specific unit or numerical magnitude. The unit, in contrast, is a standardized quantity fixed by convention, used to assign measurable values to a dimension. For example, length is a dimension, while the metre is its SI unit.

Exam Note: Dimensional Consistency and Error Detection

An equation in physics is dimensionally consistent if every additive term has the same dimensional formula. Dimensional analysis is essential in verifying the plausibility of relations, deriving new equations by equating dimensions, and identifying errors arising from incorrect combination of physical quantities in calculations. For a detailed extension of these principles in problem-solving, refer to Introduction To Dimensions.

Summary of Standard Dimensional Formulae and Their Examination Use

All measurable mechanical, electromagnetic, and thermodynamic quantities can be represented as products of integer or fractional powers of the selected base quantities. Mastery of identifying and manipulating dimensional formulae is fundamental for analysing unit conversions, establishing physical relations, and checking the correctness of equations in mechanics and allied topics. Explicit understanding and computation of dimensional and derived units is essential for high-level performance in JEE Main Mathematics and related assessments.