# Dimensions

All the physical quantities can be expressed in terms of some combination of seven fundamental (or base) quantities viz. length $[L]$, mass $[M]$, time $[T]$, electric current $[A]$, temperature $[K]$, luminous intensity $[cd]$ and amount of substance $[mol]$. The fundamental units of these quantities are metre, kilogram, second, ampere, kelvin, mole, and candela, respectively.

## Dimensions of Physical Quantities

The dimensions of a physical quantity are the powers to which the base quantities are raised to represent that quantity e.g., dimensions of force are $[\mathrm{MLT^{-2}}]$.

Use defining formula to get dimensions of uncommon physical quantities like coefficient of viscosity $\eta$, capacitance $C$, inductance $L$, surface tension $T$, electrical conductivity $\sigma$, electrical resistivity $\rho$ etc. Many physical quantities are related to energy e.g., gravitational potential energy ${GMm}/{r}$, electrostatic energy ${q_1 q_2}/{(4\pi\epsilon_0 r)}$, thermal energy $kT$ etc.

There are many dimensionless quantities like angle, strain, Poisson's ratio, refractive index, Reynold's number, relative density, coefficient of friction etc. The arguments of trigonometric, exponential, logarithm functions etc. are dimensionless.

The time constants in electrical circuits $1/RC$, $L/R$ and $1/\sqrt{LC}$ has the dimensions of time.

The dimensions of constants like Planck's constant $h$, gravitational constant $G$, gas constant $R$, Boltzmann's constant $k$, permeability $\mu_0$, permittivity $\epsilon_0$, Stefan's constant $\sigma$, Rydberg constant $R_{\infty}$ etc.\ are frequently used.

### Quantities having same dimensions

There are many physical quantities having same dimensions:

• pressure, stress, modulus of rigidity
• surface tension, spring constant
• torque, work, energy
• impulse, momentum
• angular velocity, frequency, velocity gradient
• thermal capacity, entropy, gas constant, Boltzmann's constant
• power, luminous flux
• gravitational potential energy, latent heat etc.

## Dimensional Analysis

An equation is dimensionally correct if the dimensions of the various terms on either side of the equation are the same. This is called the principle of homogeneity of dimensions. This principle is useful in (i) checking dimensional consistency of an equation (ii) deducing relations among physical quantities, and (iii) converting a physical quantity from one system of units to another.

The dimensions of Planck's constant are same as the dimensions of angular momentum (recall quantization of angular momentum, $l=n\hbar=nh/2\pi$). The dimensional analysis gives expressions for Planck's length, Planck's mass and Planck's time in terms of fundamental constants $\hbar$, $G$ and speed of light $c$. The formula $c=1/\sqrt{\mu_0\epsilon_0}$, quickly gives dimension of $\mu_0\epsilon_0$.

By the principle of homogeneity of dimensions, if the dimensions of all the terms are not same, the equation is wrong. The dimensional analysis is also used to deduce relation among the physical quantities.

## Problems from IIT JEE

Problem (IIT JEE 2014): To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railways engineer uses dimensional analysis and assumes that the distance depends on the mass density $\rho$ of the fog, intensity (power/area) $S$ of the light from the signal and its frequency $f$. The engineer finds that $d$ is proportional to $S^{1/n}$. The value of $n$ is $\ldots$.

Solution: From the given information, \begin{align} d=k\,\rho^a S^b f^c, \end{align} where $k$ is some dimensionless proportionality constant and $a$, $b$, $c$ are unknown parameters. Substitute dimensions of physical quantities in above equation to get \begin{align} [\mathrm{L}^1]=[\mathrm{ML^{-3}}]^a [\mathrm{MT^{-3}}]^{b}[\mathrm{T}^{-1}]^c. \end{align} Equate the exponents of M and L in above equation to get \begin{align} & a+b=0,\\ &1=-3a. \end{align} Solve these equations to get $b=1/3$.