# Dimensional Formula

Before learning the dimensional formula, let us recall what is dimension. Dimension in maths is a measure of the length, width, or height extended in a particular direction. By dimension definition, it is a measure of a point or line extended in one direction. Every shape around us has some dimensions. The concept of dimension in maths does not have any specific dimensional formula. Dimension of any physical quantity is the power to which the fundamental units are raised to obtain one unit of that quantity. Let us learn about the dimensional formula with a few examples in the end.

## What Is the Dimensional Formula?

The dimensional formula of any quantity is the expression showing the powers to which the fundamental units are to be raised to obtain one unit of a derived quantity. If Q is any physical quantity, the expression representing its dimensional formula is given by,

### Dimensional Formula:

Q = M^{a}L^{b}T^{c}

where, M, L, T are base dimensions mass, length, and time respectively and a, b and, c are their respective exponents.

The following table shows dimensional formulas for different physical quantities:

Physical quantity |
Unit |
Dimensional formula |

Length | m | L |

Mass | kg | M |

Time | s | T |

Acceleration or acceleration due to gravity | ms^{–2} |
LT^{–2} |

Angle (arc/radius) | rad | M^{o}L^{o}T^{o} |

Angular displacement | rad | M^{o}L^{o}T^{o} |

Angular frequency (angular displacement/time) | rads^{–1} |
T^{–1} |

Angular impulse (torque × time) | Nms | ML^{2}T^{–1} |

Angular momentum (Iω) | kgm^{2}s^{–1} |
ML^{2}T^{–1} |

Angular velocity (angle/time) | rads^{–1} |
T^{–1} |

Area (length × breadth) | m^{2} |
L^{2} |

Boltzmann’s constant | JK^{–1} |
ML^{2}T^{–2}θ^{–1} |

Bulk modulus (ΔP × (V/ΔV)) | Nm^{–2}, Pa |
M^{1}L^{–1}T^{–2} |

Calorific value | Jkg^{–1} |
L^{2}T^{–2} |

Coefficient of linear or areal or volume expansion | ^{o}C^{–1} or K^{–1} |
θ^{–1} |

Coefficient of surface tension (force/length) | Nm^{–1} or Jm^{–2} |
MT^{–2} |

Coefficient of thermal conductivity | Wm^{–1}K^{–1} |
MLT^{–3}θ^{–1} |

Coefficient of viscosity (F = η × A × (dv/dx)) | poise | ML^{–1}T^{–1} |

Compressibility (1/bulk modulus) | Pa^{–1}, m^{2}N^{–2} |
M^{–1}LT^{2} |

Density (mass / volume) | kgm^{–3} |
ML^{–3} |

Displacement, wavelength, focal length | m | L |

Electric capacitance (charge/potential) | CV^{–1}, farad |
M^{–1}L^{–2}T^{4}I^{2} |

Electric conductance (1/resistance) | Ohm^{–1} or mho or siemen |
M^{–1}L^{–2}T^{3}I^{2} |

Electric conductivity (1/resistivity) | siemen/metre or Sm^{–1} |
M^{–1}L^{–3}T^{3}I^{2} |

Electric charge or quantity of electric charge (current × time) | coulomb | IT |

Electric current | ampere | I |

Electric dipole moment (charge × distance) | Cm | LTI |

Electric field strength or Intensity of electric field (force/charge) | NC^{–1}, Vm^{–1} |
MLT^{–3}I^{–1} |

Electric resistance (potential difference/current) | ohm | ML^{2}T^{–3}I^{–2} |

Emf (or) electric potential (work/charge) | volt | ML^{2}T^{–3}I^{–1} |

Energy (capacity to do work) | joule | ML^{2}T^{–2} |

Energy density (energy/volume) | Jm^{–3} |
ML^{–1}T^{–2} |

Entropy (ΔS = ΔQ/T) | Jθ^{–1} |
ML^{2}T^{–2}θ^{–1} |

Force (mass x acceleration) | newton (N) | MLT^{–2} |

Force constant or spring constant (force/extension) | Nm^{–1} |
MT^{–2} |

Frequency (1/period) | Hz | T^{–1} |

Gravitational potential (work/mass) | Jkg^{–1} |
L^{2}T^{–2} |

Heat (energy) | J or calorie | ML^{2}T^{–2} |

Illumination (Illuminance) | lux (lumen/metre^{2}) |
MT^{–3} |

Impulse (force x time) | Ns or kgms^{–1} |
MLT^{–1} |

Inductance (L) (energy = \(\frac{1}{2}\) LI^{2} or
Coefficient of self-induction |
henry (H) | ML^{2}T^{–2}I^{–2} |

Intensity of gravitational field (F/m) | Nkg^{–1} |
L^{1}T^{–2} |

Intensity of magnetization (I) | Am^{–1} |
L^{–1}I |

Joule’s constant or mechanical equivalent of heat | Jcal^{–1} |
M^{o}L^{o}T^{o} |

Latent heat (Q = mL) | Jkg^{–1} |
M^{o}L^{2}T^{–2} |

Linear density (mass per unit length) | kgm^{–1} |
ML^{–1} |

Luminous flux | lumen or (Js^{–1}) |
ML^{2}T^{–3} |

Magnetic dipole moment | Am^{2} |
L^{2}I |

Magnetic flux (magnetic induction x area) | weber (Wb) | ML^{2}T^{–2}I^{–1} |

Magnetic induction (F = Bil) | NI^{–1}m^{–1} or T |
MT^{–2}I^{–1} |

Magnetic pole strength | Am (ampere–meter) | LI |

Modulus of elasticity (stress/strain) | Nm^{–2}, Pa |
ML^{–1}T^{–2} |

Moment of inertia (mass × radius^{2}) |
kgm^{2} |
ML^{2} |

Momentum (mass × velocity) | kgms^{–1} |
MLT^{–1} |

Permeability of free space \(\left(μ_o = \dfrac{4\pi Fd^{2}}{m_1m_2}\right)\) | Hm^{–1} or NA^{–2} |
MLT^{–2}I^{–2} |

Permittivity of free space \(\left({{\varepsilon }_{o}}=\frac{{{Q}_{1}}{{Q}_{2}}}{4\pi F{{d}^{2}}}\right)\) | Fm^{–1} or C^{2}N^{–1}m^{–2} |
M^{–1}L^{–3}T^{4}I^{2} |

Planck’s constant (energy/frequency) | Js | ML^{2}T^{–1} |

Poisson’s ratio (lateral strain/longitudinal strain) | –– | M^{o}L^{o}T^{o} |

Power (work/time) | Js^{–1} or watt (W) |
ML^{2}T^{–3} |

Pressure (force/area) | Nm^{–2} or Pa |
ML^{–1}T^{–2} |

Pressure coefficient or volume coefficient | ^{o}C^{–1} or θ^{–1} |
θ^{–1} |

Pressure head | m | M^{o}LT^{o} |

Radioactivity | disintegrations per second | M^{o}L^{o}T^{–1} |

Ratio of specific heats | –– | M^{o}L^{o}T^{o} |

Refractive index | –– | M^{o}L^{o}T^{o} |

Resistivity or specific resistance | Ω–m | ML^{3}T^{–3}I^{–2} |

Specific conductance or conductivity (1/specific resistance) | siemen/metre or Sm^{–1} |
M^{–1}L^{–3}T^{3}I^{2} |

Specific entropy (1/entropy) | KJ^{–1} |
M^{–1}L^{–2}T^{2}θ |

Specific gravity (density of the substance/density of water) | –– | M^{o}L^{o}T^{o} |

Specific heat (Q = mst) | Jkg^{–1}θ^{–1} |
M^{o}L^{2}T^{–2}θ^{–1} |

Specific volume (1/density) | m^{3}kg^{–1} |
M^{–1}L^{3} |

Speed (distance/time) | ms^{–1} |
LT^{–1} |

Stefan’s constant \(\left( \frac{\text{heat energy}}{\text{area} \times \text{time} \times \text{temperature}^{4}} \right)\) | Wm^{–2}θ^{–4} |
ML^{o}T^{–3}θ^{–4} |

Strain (change in dimension/original dimension) | –– | M^{o}L^{o}T^{o} |

Stress (restoring force/area) | Nm^{–2} or Pa |
ML^{–1}T^{–2} |

Surface energy density (energy/area) | Jm^{–2} |
MT^{–2} |

Temperature | ^{o}C or θ |
M^{o}L^{o}T^{o}θ |

Temperature gradient \(\left(\frac{\text{change in temperature}}{\text{distance}}\right)\) | ^{o}Cm^{–1} or θm^{–1} |
M^{o}L^{–1}T^{o}θ |

Thermal capacity (mass × specific heat) | Jθ^{–1} |
ML^{2}T^{–2}θ^{–1} |

Time period | second | T |

Torque or moment of force (force × distance) | Nm | ML^{2}T^{–2} |

Universal gas constant (work/temperature) | Jmol^{–1}θ^{–1} |
ML^{2}T^{–2}θ^{–1} |

Universal gravitational constant \(\left(F = G. \frac{{{m}_{1}}{{m}_{2}}}{{{d}^{2}}}\right)\) | Nm^{2}kg^{–2} |
M^{–1}L^{3}T^{–2} |

Velocity (displacement/time) | ms^{–1} |
LT^{–1} |

Velocity gradient (dv/dx) | s^{–1} |
T^{–1} |

Volume (length × breadth × height) | m^{3} |
L^{3} |

Water equivalent | kg | ML^{o}T^{o} |

Work (force × displacement) | J | ML^{2}T^{–2} |

Decay constant | s^{-1} |
M^{0}L^{0}T^{-1 } |

Potential energy | J | M^{1}L^{2}T^{-2} |

Kinetic energy | J | M^{1}L^{2}T^{-2} |

## Dimensional Formula and Dimensional Equations

A dimensional equation is an equation that relates fundamental units and derived units in terms of dimensions. In mechanics, the length, mass, time, temperature, and electric current are taken as three base dimensions, and meter, kilogram, second, ampere, kelvin, mole, and candela are the fundamental units. The dimensional formula of individual quantities is used to establish a relationship between them in any dimensional equation. An example of a dimensional equation is as given below,

Dimensional formula(equation) for area:

Area = length × breadth

= length × length

= [L] × [L]

= [L]^{2}

⇒ Dimensional formula (equation) for area (A) = [L^{2} M^{0} T^{0}]

## Applications of Dimensional Formula

The dimensional formula finds applications in the following cases,

- It is used to verify the correctness of an equation.
- The dimensional formula helps in deriving the relationship between different physical quantities.
- For conversion from one system of units into another system for any given quantity.
- It expresses one quantity in terms of the fundamental units.

Let us have a look at a few solved examples to understand the dimensional formula better.

## Examples Using Dimensional Formulas

**Example 1:** Using dimensional formula, Q = M^{a}L^{b}T^{c}, find the values of a, b, and c if the given quantity is velocity.

**Solution:**

To find: Values for a, b, and c

Given:

Quantity = Velocity

Using the dimensional formula,

Q = M^{a}L^{b}T^{c}

We know,

Velocity = (displacement/time)

= L/T

= M^{0}L^{1}T^{-1}

Comparing with dimensional formula, we get,

a = 0, b = 1, c = -1

**Answer: **a = 0, b = 1, c = -1

**Example 2:** Find the dimensional formula of momentum.

**Solution:**

To find: Dimensional formula of momentum

We know,

Momentum = (mass × velocity)

= [MLT^{-1}]

**Answer: **Dimensional formula for momentum = [MLT^{-1}]

**Example 3: **State and verify the formula for acceleration using the dimensional analysis.

**Solution:**

The formula for acceleration is given as, a = change in velocity/time taken = ∆V/∆t

Using dimensional analysis,

Acceleration = change in velocity/time taken

Dimesional formula for LHS = [LT^{–2}]

Dimesional formula for RHS = [LT^{–1}]/[T] = [LT^{–2}]

Since, LHS = RHS, the given formula is verified dimensionally.

## FAQs on Dimensional Formula

### What Is Meant By Dimensional Formula?

The expression depicting the powers to which the fundamental units are to be raised to obtain one unit of a derived quantity is known as the dimensional formula. It is given as, Q = M^{a}L^{b}T^{c}, where, M, L, T are base dimensions with respective exponents a, b and, c. and Q is the physical quantity.

### How do you Find the Dimensional Formula?

The dimensional formula of any quantity can be given by expressing the formula for it and breaking it down in terms of the base dimensions. Using these base dimensions, we can evaluate the dimensional formula for any given quantity.

### What Is Dimensional Formula of Frequency?

The dimensional formula for frequency is given as, [MT^{–2}]. The unit for frequency is hertz.

### What Are the Uses of Dimensional Formula?

The dimensional formula is used to verify the correctness of an equation and it helps in deriving the relationship between different physical quantities. For conversion of one system of units into another system for any given quantity, we follow the dimensional analysis.