# Dimension of physical quantities, Uses & Limitations

Dimension of physical quantities is defined as the power to which the fundamental units are raised to represent it. The fundamental units of mass i.e kilogram is expressed in dimensions as [M], length as [L], time as [T], Temperature as [K] and so on. Dimensions of physical quantities are mainly calculated for derived units.

## List of dimensions of physical quantities

List of dimensions of physical quantities with formula
S.N Physical quantity Formula Dimensional Formula Unit
1 Force m x a [MLT-2] Newton(N)
2 Thrust v x dm/dt
3 Weight m x g
4 Tension mg ± f
5 Energy Gradient E/L N/m
6 Work F x d [ML2T-2] Kgm2s-2 or Joule
7 Energy F x d Kgm2s-2 or Joule
8 Torque F x r Nm
9 Couple of force F x s
10 Moment of force F x d
11 Internal Energy Q - W Joule
12 Heat msdt Joule
13 Work function hf - E Joule or ev
14 Spring constant F/x [ML0T-2] N/m
15 Force constant F/x
16 Surface tension F/L
17 Surface Energy Work/Area
18 Acceleration V/t [M0LT-2] m/s2
19 Gravitational Field strength Force/mass N/kg
20 Acceleration due to gravity(g) Gm/r2 m/s2
21 Angular velocity Angle/time [T-1] radian/second
22 Frequency 1/T s-1
23 Veloity gradient ∆v/L
24 Radioactive Decay Constant(λ) 0.693/T1/2
25 Hubble’s Constant(Hₒ) velocity/distance
26 Linear momentum M x v [MLT-1] Kgms-1 or Ns
27 Impulse F x t
28 Angular momentum I x ω [ML2T-1] Js
29 Plannk's constant E/f
30 Pressure F/A [ML-1T-2] Kgm-1s-2 or N/m2
31 Stress F/A
32 Modulus of Elasticity(Young’s & Bulk modulus) Stress/strain
33 Energy Density E/A
34 Gravitatiional Potential Work/mass [M0L2T-2] J/kg
35 Specific Latent Heat Q/m
36 Thermal capacity ∆Q/∆T [ML2T-2K-1] J/Kelvin
37 Boltzmann's Constant R/N
38 Entropy ∆H/T
39 Specific Heat Capacity(s) Q/mdt [M0L2T-2K-1] Jkg-1K-1
40 Universal Gas Constant(R) PV/nT J/mole Kelvin
41 Intensity of Radiation ɸ/solid angle [ML0T-3] Watt/steradian
42 Solar Constant E/A x t Watt/m2
43 Magnetizing Force B/µ [M0L-1T0A] Ampere/m
44 Intensity of Magnetization M/V
45 Electric potential
Work done/charge
[ML2T3A-1] volt
46 EMF
47 Electric Potential Difference
48 Rydberg's Constant(R) 1/ λ=R(1/n12-1/n22) [M0L-1T0] m-1
49 Wave Number 1/λ
50 Magnetic Moment(M) NIA [AL2] Am2
51 Bohr’s Magneton(µ) neh/4πme Joule/Tesla
52 Permitivity(𝜀ₒ) Fq1q2/4π 𝜀ₒr2 [MLT-2A-2] Farad/m
53 Permeability(µₒ) B/H Henry/m
54 Dielectric Constant(ĸ) C/Cₒ Dimensionless Unitless
55 Relative Permitivity(𝜀r) 𝜀/𝜀ₒ
56 Relative Permeability(µr) µ/µₒ

## Uses of Dimensions

1. It is used for conversion of one system of unit into another system of unit. Example: conversion of numerical value in S.I unit to C.G.S unit.
2. It is used for checking the result. This depends on the principle of homogeneity of dimensions. According to this principle, the dimension of all terms on left hand side of equation is always equal to the dimension of all terms on right hand side of the equation.

For example,
s = ut + 1/2gt2
Thedimension of displacement (s) must be equal to the dimension on right hand side of the equation. This equation is dimensionally correct because on calculating dimension of right hand side terms, it is equal to [L], same as left hand term if we calculate.
3. It is used for deriving a result.

## How to calculate dimension of any physical quantities

Step 1

To, calculate dimension, we first need to know the formula of that quantities. For examples, formula of force is given by,

F = mass(m) x acceleration(a)

Step 2

Next, we should simplify the formula until all the terms in that formula changed into fundamental quantity. This step is need because only fundamental quantities are expressed in dimensional form.

In above formula of force, mass is fundamental quantity and can be represented as [M]. But acceleration is not fundamental quantity and needs to be simplified further as,

F = mass(m) x acceleration(a)

F = m x v/t  [a=v/t]

F = m x d/t2  [v=d/t]

Step 3

Finally, all the terms are expressed in respective dimensions in capitalized form. The power if present are raised to respective dimensions. The opening and closing big bracket [] is used to enclose all units.

In above formula of force,

F = m x d/t2

F = [MLT-2]

We can also find the dimension from its fundamental unit. For example unit of force is kgms-2. The dimensional formula can be derived as [M] for kg, [L] for m and [T-2] for s-2 which makes [MLT-2].

## Limitation of Dimensions

• It is not applicable for trigonometric and exponential functions.
• It always does not define the exact form of a relation.
• A dimensionally correct equation may or may not be physically correct equation.

## Calculating dimensions of some quantities

Dimension of speed /velocity

V = d/t

V = L/T

V = [LT-1]

Dimension of Work or Energy

E = F x d

= m x a x d

= m x v /t x d

= m x d /t2 x d

= m x d2/t2

= [ML2T-2]

Dimension of linear momentum = m x v

= m x d/t

= [MLT-1]

Dimension of modulus of rigidity

Dimension of angular momentum = Inertia(I) x ω

= m x r2 x 1/t

=[ML2T-1]

Dimension of universal gas constant(R)

From equation,

PV = nRT

R = PV/nT

R = E/nT {since PV is equal to work done or energy, E}

R = [ML2T-2mole-1K-1]

Dimension of Boltzmann’s constant (K)

K = internal energy/Temperature

From above dimension of energy is calculated as [ML2T-2]

K = [ML2T-2] / [K]

K = [ML2T-2K-1]

Dimension of Planck's constant

Dimension of magnetic moment (M)

M = NIA {where N= no of turns=constant}

M = IA

M = [AL2]

Dimension of entropy(S)

S = ?Q/Temp

S = [ML2T-2] / [K]

S = [ML-2T-2K-1]

Dimension of permitivity(𝜀ₒ)

From equation,

F = 1/4π 𝜀ₒ q1q2/r2

On cross multiplication

𝜀ₒ = q1q2/Fr2 {1/4π is constant and do not have dimension so removed}

= It x It/Fr2 {since current, I = q/t}

= I2t2 / Fr2

= [A2T2] / [MLT-2] x [L2]

= [A2T2] / [ML3T-2]

= [M-1L-3T4A2]

## Dimensionless quantities

They are the quantities which does not have dimensions. So, they are called as dimensionless quantities. A dimensionless quantity also does not have unit. Some of them are:

Poisson’s ratio

Strain

Angle & solid angle

Refractive index

Relative density

Emissivity

Dielectric constant

Refractive index

Coefficient of friction

Relative permeability

Relative permittivity

Magnetic susceptibility

Electric susceptibility

## Key Points

A physical quantity may be dimensionless but may have units. For example, plane angle has unit radian but is dimensionless.

A unitless quantity never have dimensions.

Example of dimensional constant are: gravitational constant, plank’s constant, Stefan’s constant, universal gas constant etc.

Examples of non-dimensional constant are: pure numbers (1,2,3,4,…..) and all trigonometric functions.

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