Definition

The dimension of a physical quantity is the power (or exponents) to which the base quantities are raised to represent that quantity. In other words, the expression for physical quantity in terms of the base quantities is called the dimensional formula. This formula shows how and which basic quantities are involved in given physical quantities.

For convenience, the base quantities are represented by one letter symbol. Generally, mass is denoted by M, length by L and the time by T, and electric current by A or I. The thermodynamic temperature, amount of substance, and luminous intensity are denoted by the symbol of their units i.e., K, mol, and cd respectively. These base quantities are always enclosed in square brackets [] to note that the equation is dimensional and not among the magnitude.

In mechanics, all physical quantities can be written in terms of the dimensions [L], [M], and [T]. For example, the volume occupied by an object is expressed as the product of length, breadth, and height. Hence the dimension of volume are [L] × [L] × [L] = [L]³. As the volume is independent of mass and time, it is said to possess zero dimension in Mass [M⁰] and zero dimension in time [T⁰], and three dimensions in length.

For example:

The dimensional formula of force is [M L T^-2],
The dimensional formula of momentum is [M L T^-1], and
The dimensional formula of speed is [M⁰ L T^-1]

Dimensional Formula Vs Equations

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 volume is [M⁰ L³ T⁰]. Whereas, an equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity. for example, [V] = [M⁰ L³ T⁰]. Other examples of dimensional equations are

The dimensional equation of volume, [V] = [M⁰ L³ T⁰],

The dimensional equation of speed [ν] = [M⁰ L T^-1],

The dimensional equation of force [F] = [M L T^-2], and

The dimensional equation of density [ρ] = [M L^-3 T⁰].

The dimensional equation can be obtained from the equation representing the relations between the physical quantities.

Principle of Homogenity of Dimensions

According to this principle, for the physical relation to be correct, the dimensions of the fundamental quantities on the left-hand side of the equation must be equal to the dimensions of the fundamental quantities on the right-hand side of the equation.”

For illustration, we consider an equation A = B + C. For this equation to be correct,
dimensions of B and C must be equal to dimensions of A.

Dimensional Formula of Physical Quantities

The following table contains commonly used physical quantities and their dimensional formulas.

S.N.	Physical Quantity	Physical Formula	Dimensions	Dimensional Formula
1.	Area	Length × breadth	[L2]	[M0 L2 T0]
2.	Volume	length × breadth × width	[L3]	[M0 L3 T0]
3.	Mass density	Mass / Volume	[M]/[L3] or [M L-3]	[M L-3 T0]
4.	Frequency	1 / Time period	1 / [T]	[M0 L0 T-1]
5.	Velocity, speed	displacement / time	[L]/[T]	[M0 L T-1]
6.	Acceleration	velocity / time	[L T-1]/[T]	[M0 L T-2]
7.	Force	Mass × acceleration	[M][L T-2]	[M L T-2]
8.	Impulse	Force × time	[M L T-2][T]	[M L T-1]
9.	Work, Energy	Force × distance	[M L T-2][L]	[M L2 T-2]
10.	Power	Work/time	[M L2 T-2]/[T]	[M L2 T-3]
11.	Momentum	Mass × velocity	[M L T-2]/[L2]	[M L T-1]

• Dimension of a physical quantity is the power to which the fundamental units must be raised, in order to represent it.
  
  
• The dimension of mass is denoted as [M],
  the dimension of length is denoted as [L],
  the dimension of time is denoted as [T],
  the dimension of temperature is denoted as [K],
  the dimension of electric current is denoted as [A],
  the dimension of luminous intensity is denoted as [cd] and
  the dimension of substance is denoted as [mol]
  
  
• The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula
  
  
• The dimensional equation can be obtained from the equation representing the relations between the physical quantities.
  
  
• Angle is a dimensionless physical quantity.