Dimensional Formula

Dimensional Formulas play an important role in converting units from one system to another and find numerous practical applications in real-life situations. Dimensional Formulas are a fundamental component of the field of units and measurements. In mathematics, Dimension refers to the measurement of an object’s size, extent, or distance in a specific direction, such as length, width, or height, but in the context of physical quantities, the dimension signifies the exponent to which fundamental units must be raised to yield a single unit of that specific quantity.

In this article, we will discuss the introduction, definition, properties, and limitations of a Dimensional Formula and its meaning. We will also understand dimensional formulas for different physical quantities and Dimensional equations. We will also solve various examples and provide practice questions for a better understanding of the concept of this article. We have to study Dimensional Formula in Class 11.

What is Dimensional Formula?

The Dimensional Formula of any quantity serves as an expression that shows the powers by which fundamental units must be raised to yield a single unit of that derived quantity. These dimensional formulas play an important role in establishing relationships between variables in nearly every dimensional equation.

These formulae, also known as the Dimensional Formula of the Physical Quantity, tell us about the presence and combination of fundamental quantities within a given physical quantity. A dimensional formula is always enclosed within square brackets [ ].

Example of Dimensional Formula

Let’s suppose there is a physical quantity X that depends on the fundamental dimensions of Mass (M), Length (L), and Time (T), each with associated powers a, b, and c, then its Dimensional Formula can be expressed as follows:

Dimensional Formulae X = [M^aL^bT^c]

Dimensional Formula for Various Quantities

The table below provides Dimensional Formulas for different physical quantities:

Physical Quantity	Unit	Dimensional Formula
Acceleration or Acceleration due to Gravity	ms-2	LT-2
Angular Displacement	rad	M0L0T0
Angular Impulse	Nms	ML2T-1
Angular Velocity (angle/time)	rads-1	T-1
Angle (Arc/Radius)	rad	M0L0T0
Angular Frequency (Angular Displacement/Time)	rads-1	T-1
Angular Momentum	kgm2s-1	ML2T-1
Boltzmann’s Constant	JK-1	ML2T-2θ–
Bulk Modulus	N/m2	ML-1T-2
Calorific Value	JKg-1	L2T-2
Coefficient of Surface Tension (Force/Length)	N/m	MT-2
Coefficient of Linear or Areal or Volume Expansion	K-1	θ-1
Coefficient of Thermal Conductivity	Wm-1K-1	MLT-3θ-1
Compressibility (1/Bulk Modulus)	m2N-2	M-1LT2
Density (Mass / Volume)	Kg/m3	ML-3
Displacement, Wavelength, Focal Length	m	L
Electric Capacitance (Charge/Potential)	farad	M-1L-2T4I2
Electric Conductivity (1/Resistivity)	Sm-1	M-1L-3T3I2
Electric Current	ampere	I
Electric Field Strength or Intensity of Electric Field (Force/Charge)	NC-1	MLT-3I-1
Emf (or) Electric Potential (Work/Charge)	volt	ML2T-3I-1
Energy Density (Energy/Volume)	Jm-3	ML-1T-2
Electric Conductance (1/Resistance)	Ohm-1	ML-1T-2T3I2
Electric Charge or Quantity of Electric Charge	coulomb	IT
Electric Dipole Moment	Cm	LTI
Electric Resistance (Potential Difference/Current)	ohm	ML2T-3I-2
Energy (Capacity to do work)	joule	ML2T-2
Entropy	Jθ–1	ML2T-2θ-1
Force	newton (N)	MLT-2
Frequency (1/period)	Hz	T-1
Force Constant or Spring Constant (Force/Extension)	Nm-1	MT-2
Gravitational Potential (Work/Mass)	J/kg	L2T-2
Heat (Energy)	J or calorie	ML2T-2
Illumination (Illuminance)	lumen/m2	MT-3
Inductance	henry (H)	ML2T-2I-2
Intensity of Magnetization (I)	Am-1	L-1I
Impulse	Ns	MLT-1
Intensity of Gravitational Field (F/m)	Nkg-1	LT-2
Joule’s Constant	Jcal-1	M0L0To
Latent Heat (Q = mL)	Jkg-1	L2T-2
Luminous Flux	Js-1	ML2T-3
Linear density (mass per unit length)	Kgm-1	ML-1
Magnetic Dipole Moment	Am2	L2I
Magnetic induction (F = Bil)	NI-1m-1	MT-2I-1
Modulus of Elasticity (Stress/Strain)	Pa	ML-1T-2
Momentum	kgms-1	MLT-1
Magnetic Flux	weber (Wb)	ML2T-2I-1
Magnetic Pole Strength	Am (ampere–meter)	LI
Moment of Inertia	Kgm2	ML2
Planck’s Constant (Energy/Frequency)	Js	ML2T-1
Power (Work/Time)	watt (W)	ML2T-3
Pressure Coefficient or Volume Coefficient	θ-1	θ-1
Permittivity of Free Space	Fm-1	M-1L-3T4I2
Poisson’s Ratio (Lateral Strain/Longitudinal Strain)	Dimensionless	M0L0T0
Pressure (Force/Area)	N/m2	ML-1T-2
Pressure Head	m	L
Radioactivity	disintegrations per second	T-1
Refractive Index	Dimensionless	M0L0T0
Specific Conductance or Conductivity (1/Specific Resistance)	Sm-1	M-1L-3T3I2
Specific Gravity (Density of the Substance/Density of Water)	Dimensionless	M0L0T0
Specific Volume (1/Density)	m3kg-1	M-1L3
Stress (Restoring Force/Area)	N/m2	ML-1T-2
Ratio of Specific Heats	Dimensionless	M0L0T0
Resistivity or Specific Resistance	Ω-m	ML3T-3I-2
Specific Entropy (1/entropy)	KJ-1	M-1L-2T2θ
Specific Heat (Q = mst)		L2T-2θ-1
Speed (Distance/Time)	m/s	LT-1
Strain (Change in Dimension/Original dimension)	Dimensionless	M0L0T0
Surface Energy Density (Energy/Area)	J/m2	MT-2
Temperature	θ	θ
Thermal Capacity	Jθ-1	ML2T-2θ-1
Torque or Moment of Force	Nm	ML2T-2
Temperature Gradient	θm-1	L-1θ
Time Period	second	T
Universal Gas Constant (Work/Temperature)	Jmol–1θ-1	ML2T-2θ-1
Universal Gravitational Constant	Nm2kg-2	M-1L3T-2
Velocity (Displacement/Time)	m/s	LT-1
Volume	m3	L3
Velocity Gradient (dv/dx)	s-1	T-1
Water Equivalent	kg	M
Work	J	ML2T-2
Decay Constant	s-1	T-1
Kinetic Energy	J	ML2T-2
Potential Energy	J	ML2T-2

Application of Dimensional Formula

Some of the common applications of dimensional formula are:

• To verify whether a formula is dimensionally correct or not.
• Conversion of units from one system to another for any given quantity.
• To establish derivation between physical quantities based on mutual relationships.
• Dimensional Formulae express every physical quantity in terms of fundamental units.

Limitations of Dimensional Formula

While Dimensional Formulas offer numerous benefits, they also come with certain limitations:

• It’s important to note that quantities like trigonometric functions, plane angles, and solid angles do not possess defined dimensional formulae since they are inherently dimensionless in nature.
• The applicability of dimensional formulas is confined to a specific set of physical quantities.
• They are unable to determine proportionality constants, which can be a drawback in certain situations.
• Dimensional Formulas are primarily suitable for addition and subtraction operations, limiting their use in other mathematical operations.

Dimensional Formula and Dimensional Equations

The equations resulting from equating a physical quantity to its dimensional formula are termed Dimensional Equations. These equations are an important tool for representing physical quantities in terms of fundamental units. Dimensional formulas for specific quantities used as a foundation for establishing relationships between those quantities within any given dimensional equation.

For example, consider a physical quantity denoted as Y, which depends on the fundamental quantities M (mass), L (length), and T (time) with respective powers a, b, and c. The dimensional formula for this physical quantity [Y] can be expressed as:

[Y] = [M^aL^bT^c]

As examples:

• The dimensional equation for velocity ‘v’ is expressed as [v] = [M⁰L¹T^-1].
• The dimensional equation for acceleration ‘a’ is denoted as [a] = [M⁰L¹T^-2].
• The dimensional equation for force ‘F’ is given as [F] = [M¹L¹T^-2].
• The dimensional equation for energy ‘E’ is represented as [E] = [M¹L²T^-2].
• These dimensional equations provide a way to understand and represent various physical quantities in terms of their fundamental units.

Read More,

• List of SI Units in Measurement
• System of Units
• Dimensional Analysis

Solved Examples on Dimensional Formula

Example 1: Using Dimensional Formula, X= M^aL^bT^c, find the values of a, b, and c for density.

Solution:   

To find: Values for a, b, and c

Given:

Quantity = Density

Using the Dimensional Formula,

X = M^aL^bT^c

We know,

Density = (mass/length³)

= M/L³

= M¹L^-3T⁰

Comparing with Dimensional Formula, we get,

a = 1, b = -3, c = 0

Answer: a = 1, b = -3, c = 0

Example 2: Determine the Dimensional Formula of velocity.

Solution:     

To find: Dimensional formula of velocity

We know,

Velocity = (distance/time)

= [M⁰L¹T^-1]

Answer: Dimensional formula for velocity = [M⁰L¹T^-1]

Example 3: State and verify the formula for pressure using the Dimensional Formula analysis.

Solution: 

The formula for Pressure is given as, P = Force/Area= F/A

Using Dimensional Formula analysis,

Pressure =  Force/Area
Dimesional formula for LHS = [M¹L^-1T^–2]
Dimesional formula for RHS = [M¹L¹T^–2]/[L²] = [M¹L^-1T^–2] 
Since LHS matches RHS, the given formula for Pressure is verified dimensionally.

Practice Questions on Dimensional Formula

Q1. Using Dimensional Formula, X= M^aL^bT^c, find the values of a, b, and c for Energy.

Q2. Using Dimensional Formula, X= M^aL^bT^c, find the values of a, b, and c for Acceleration.

Q3. Determine the Dimensional Formula of Power.

Q4. Determine the Dimensional Formula of Time period of wave.

Dimensional Formula – FAQs

1. What Is Meant By Dimensional Formula?

The Dimensional Formula of any quantity serves as an expression that shows the powers by which fundamental units must be raised to yield a single unit of that derived quantity. These Dimensional Formula play an important role in establishing relationships between variables in nearly every dimensional equation.

2. 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.

3. Write down the Dimensional Formula of Frequency.

The Dimensional Formula for frequency is [MT^–2]. Hertz is the unit of frequency. 

4. What Are the Uses of Dimensional Formula?

• To verify whether a formula is dimensionally correct or not.
• Conversion of units from one system to another for any given quantity.
• To establish derivation between physical quantities based on mutual relationships.
• Dimensional Formulae express every physical quantity in terms of fundamental units.

5. Write down the difference between Dimensional Quantities and Dimensional Formula?

The equations resulting from equating a physical quantity to its dimensional formula are termed Dimensional Equations. These equations are an important tool for representing physical quantities in terms of fundamental units. Dimensional formulas for specific quantities used as a foundation for establishing relationships between those quantities within any given dimensional equation.

6. What is Dimension in mathematics?

In mathematics, Dimension refers to the measurement of an object’s size, extent, or distance in a specific direction, such as length, width, or height. As per the definition of Dimension, it represents the extent of a point or a line in a specific direction.

7. What are the Important Dimensional Formula?

Length = L Mass = M Time are the important Dimensional Formula.

8. Write down the Limitation of Dimensional Formula.

• It’s important to note that quantities like trigonometric functions, plane angles, and solid angles do not possess defined dimensional formulae since they are inherently dimensionless in nature.
• The applicability of Dimensional Formulas is confined to a specific set of physical quantities.
• They are unable to determine proportionality constants, which can be a drawback in certain situations.
• Dimensional formulas are primarily suitable for addition and subtraction operations, limiting their use in other mathematical operations.