Dimensions of Physical Quantities

The Idea of Dimensions

Every physical quantity has a nature — it is not just a number, but a kind of thing. Length, mass, and time are fundamentally different from each other. This "kind" is what we call its dimension. When we say a quantity has the dimension of length, we mean it is the same type of quantity as length, regardless of the unit we use to measure it (metres, feet, light-years).

All physical quantities can be expressed as combinations of a small set of base quantities. In physics, we recognise seven such base quantities, and we treat each as a fundamental dimension. These are:

• Length: $[L]$
• Mass: $[M]$
• Time: $[T]$
• Electric current: $[A]$
• Thermodynamic temperature: $[K]$
• Luminous intensity: $[cd]$
• Amount of substance: $[mol]$

The square brackets $[ \ ]$ are a notation that means "the dimensions of". So $[L]$ is read as "the dimension of length".

The dimensions of a physical quantity are the powers (or exponents) to which these base quantities are raised to represent that quantity. For example, speed has dimensions of length divided by time. Its dimensional formula is $[L T^{-1}]$. This tells us that speed has one dimension in length, a dimension of $-1$ in time, and zero dimensions in mass, current, temperature, and so on.

Dimensional Formulae of Common Quantities

Let's see how to find the dimensions of a quantity from its defining equation.

Volume is defined as length × breadth × height. Each of these is a length.

Volume has three dimensions in length, and zero dimensions in mass $[M^0]$ and time $[T^0]$.

Force is defined by Newton's second law: Force = mass × acceleration.

Acceleration is change in velocity per unit time. Velocity is length per unit time, $[L T^{-1}]$. Therefore:

Now, force:

Force has one dimension in mass, one in length, and $-2$ in time. …