Dimensions Of Physical Quantities

Dimensions Of Physical Quantities

Edited By Vishal kumar | Updated on Jul 02, 2025 06:51 PM IST

Physical quantities are the measurable aspects of our world, describing everything from the speed of a car to the brightness of a star. Each physical quantity is associated with a dimension, such as length, mass, or time, which helps us understand and quantify the universe around us. For instance, consider the dimensions of force, which involve mass, length, and time—this is what allows engineers to design safe bridges or vehicles that can withstand different forces. Similarly, understanding the dimensions of physical quantities is crucial in fields like medicine, where precise measurements can mean the difference between a successful treatment and a failed one. By grasping these fundamental concepts, we can better comprehend the principles that govern everything from the smallest particles to the vastness of space.

This Story also Contains
  1. Dimensions of Physical Quantities
  2. Frequency, Angular Frequency, Angular Velocity, Velocity Gradient
  3. Work, Potential Energy, Kinetic Energy, Torque
  4. Momentum, Impulse, Angular Momentum, Angular Impulse
  5. Heat, Latent heat, Specific Heat capacity and Temperature
  6. Vander Waals Constant (a and b)
  7. Voltage, Resistance and Resistivity
  8. Permittivity of Free Space and Dielectric Constant (k)
  9. Magnetic Field, Permeability of Free Space, Magnetic Flux and Self-Inductance
  10. Solved Examples Based on Dimensions of Physical Quantities
  11. Summary

Dimensions of Physical Quantities

Physical quantities are the measurable aspects of our world, describing everything from the speed of a car to the brightness of a star. Each physical quantity is associated with a dimension, such as length, mass, or time, which helps us understand and quantify the universe around us. For instance, consider the dimensions of force, which involve mass, length, and time—this is what allows engineers to design safe bridges or vehicles that can withstand different forces.

Let's now discuss Important points one by one

Dimension

The dimension of physical quantity may be defined as the power to which fundamental quantities must be raised in order to express the given physical quantities.

For representing dimensions of different quantities, we use the following symbols:

Mass - M

Length - L

Time - T

Electric current - A

Temperature - K

Amount of substance- mol

Luminous intensity - cd

Frequency, Angular Frequency, Angular Velocity, Velocity Gradient

All these quantities will have the same dimensional formula which is equal to M0L0T1

While the SI unit of Frequency and velocity gradient is sec1,

The SI unit of angular frequency and angular velocity is radians per sec

Note:- Angle is a dimensionless quantity

Work, Potential Energy, Kinetic Energy, Torque

All these quantities will have the same dimensional formula which is equal to L2T2

All these quantities will have the same unit in the SI system which is equal to N-m or Joule.

Momentum, Impulse, Angular Momentum, Angular Impulse

Momentum and Impulse both have the same dimensional formula which is equal to MLT1

Both have the same SI unit which is equal to kgms1

Angular Momentum and Angular Impulse have the same dimensional formula which is equal to ML2T1 and have the same SI unit which is equal to kg(m)2(sec)1

Heat, Latent heat, Specific Heat capacity and Temperature

Temperature

It is a fundamental quantity.

  • Dimensional formula- M0L0T0K1 (where K represents Kelvin)
  • SI unit- Kelvin

Heat

  • Dimensional formula- ML2T2

  • SI unit- Joule

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Latent Heat

Its dimensional formula is equal to M0L2T2

And its SI unit is equal to m2s2 or J/kg

Specific Heat Capacity

Dimensional formula M0L2T2K1

SI unit- JkgK

Surface Tension, Surface Energy

Surface tension- Dimensional formula- M1L0T2

SI unit- kgs2

Surface energy(per unit area) - Dimensional formula- M1L0T2

SI unit- kgs2

But (Surface tension) and (surface energy per unit area) will have the same dimensional formula and SI units.

Vander Waals Constant (a and b)

The real gas equation is given as

(P+n2aV2)(Vnb)=nRT

Where a and b are called Vander Waal's constant.

1) Vander Waal's constant (a)

Dimension- ML5T2
Unit- Newton m4

2) Vander waal 's constant (b)

Dimension- M0L3T0
Unit- m3

Voltage, Resistance and Resistivity

Voltage (V)

Dimension- ML2T3A1
Unit- Volt

Resistance (R)

Dimension- ML2T3A2
Unit- Ohm

Resistivity (ρ)

Dimension- ML3T3A2

Unit- Ohm - meter

Permittivity of Free Space and Dielectric Constant (k)

The permittivity of free space (ϵo)

Dimension- M1L3T4A2
Unit- C2N1m2 or farad/metre

Dielectric constant (k)

Dimension- M0L0T0

Unit- Unitless

Magnetic Field, Permeability of Free Space, Magnetic Flux and Self-Inductance

Magnetic Field (B)

Dimension- M1L0T2A1
Unit- newton ampere metre or volt second metre 2

Permeability of free space

The dimension of permeability of free space (μo)M1L1T2A2

SI unit- Newton/ampere2

Magnetic flux (φ)

Dimension- ML2T2A1

Unit- Weber or Volt-second

Coefficient of self-induction (L)

Dimension- ML2T2A2

Unit- Henry

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Solved Examples Based on Dimensions of Physical Quantities

Example 1: Which one of the following represents the correct dimensions of the coefficient of viscosity?

1) ML-1T-2

2) MLT-1

3) ML-1T -1

4) ML-2T-2

Solution:

Viscous force η = F * L / (A * V)
η = F / (L* V)

where F is the force acting on the fluid, η is the coefficient of viscosity, L is the length of the fluid layer, A is the area of the fluid layer, and V is the velocity of the fluid.

η=F/(L V)[η]=[MLT2][L][LT1][η]=[ML1T1]

Hence, the correct answer is the option (3).

Example 2: Out of the following pairs which one does not have identical dimensions is

1) moment of inertia and moment of force

2)work and torque

3)angular momentum and Planck’s constant

4)impulse and momentum

Solution:

Dimension of Work, Potential Energy, Kinetic Energy, Torque is ML2T2 and that of Momentum and Impulse - MLT1 and that of angular momentum \& Plank's Constant ( h ) is - M1L2T1

Moment of inertia is defined as (I)= Mass (M)xradius2(r)
The dimensional formula is =ML2
Know that the moment of force, T= radius (r)x force (F)
The dimensional formula for the moment of a force is- ML2 T2

Therefore the dimension of a moment of inertia and a moment of force does not have an identical dimension and torque is also called a moment of force.

Example 3: Given below are two statements :
Statements (I): Planck’s constant and angular momentum have the same dimensions
Statements (II): Linear momentum and moment of force have the same dimensions
In light of the above statements choose the correct answer from the option given below

1)Statement (I) is true but Statement II is false

2) Statement (I) is false but Statement II is true

3)Both statement I and statement II are true
4)Both statement I and statement II are false

Solution:

(1) L=n2πh

Statement I is true.

(2) p=mv[p]=[MLT1]τ=rF[τ]=[ML2T2]

Statement II is false.

Hence, the answer is the option (1).

Example 4: The dimensional formula of heat (Q) is

1) [M0L2T2]
2) [M1L2T2]
3) [M2L1T2]
4) [M1L2T1]

Solution:

Heat is a form of energy. Hence, The dimensions of heat must be equal to that of energy or work done.

Work done
W=Fd
Dimensions of work done or heat is

=[M1L1T2][L1][ML2T2]

Hence, the answer is the option (2).

Example 5: The dimensional formula [M0L2T2] is equal to the dimensional formula of

1) Electric Potential

2) Gravitational potential

3) Latent heat

4) Both B and C

Solution:

Latent heat and gravitational potential

M0L2T2 wherein dimension- m2s2 Electrical Potential =Jc[V]=[ML2T2]AT=[ML2T3A] Gravitational potential = Potential Energy mass =[ML2T2]M=[M0L2T2] Latent Heat (L) Qm[L]=[ML2T2]M=[L2T2]

Hence, the answer is the option (4).

Summary

Dimensions of physical quantities The nature of a PHYSICAL QUANTITY is expressed in terms of fundamental quantities such as length L, mass M, and time T. For example, the dimensions for speed are length per time, L/T, whereas for force, it is mass times acceleration, ML/T². These types of dimensions help us make sense of physical relationships between various variables and keep the equations consistent. With this analysis, we could check the rightness of equations, change units, and even solve very complicated problems. Understanding the dimensions of a physical quantity is an exceedingly important area of acquiring accurate measurements, prosperous scientific experimentation, and practical utility in all life pursuits.

Frequently Asked Questions (FAQs)

1. What are dimensions in physics?
Dimensions in physics are the fundamental physical quantities that describe a measurement. They represent the nature of a physical quantity independent of the units used. The basic dimensions are length (L), mass (M), time (T), temperature (θ), electric current (I), luminous intensity (J), and amount of substance (N).
2. How do dimensions differ from units?
Dimensions represent the nature of a physical quantity, while units are specific measures used to quantify that quantity. For example, length has the dimension [L], but can be measured in units like meters, feet, or kilometers. Dimensions are more fundamental and don't change with unit systems.
3. Why are dimensions important in physics?
Dimensions are crucial in physics for several reasons: they help in checking the correctness of equations (dimensional analysis), aid in deriving new equations, and allow for the conversion between different unit systems. They provide a fundamental understanding of physical quantities beyond specific measurement systems.
4. What is dimensional homogeneity?
Dimensional homogeneity is the principle that states that all terms in a physical equation must have the same dimensions. This ensures that the equation is physically meaningful and consistent. If an equation is not dimensionally homogeneous, it indicates an error in the formulation or understanding of the physical relationship.
5. Can you add or subtract quantities with different dimensions?
No, you cannot add or subtract quantities with different dimensions. Only quantities with the same dimensions can be added or subtracted. For example, you can add lengths to lengths, but you cannot add length to mass. This is a fundamental rule in physics that ensures physical consistency.
6. What is the dimension of velocity?
The dimension of velocity is [LT^-1]. This is because velocity is defined as distance (length) divided by time. The dimension of length [L] divided by the dimension of time [T] gives [LT^-1].
7. What is the dimension of angular velocity?
The dimension of angular velocity is [T^-1]. This is because angular velocity is defined as the rate of change of angle with respect to time, and angle is a dimensionless quantity. This dimension helps in analyzing rotational motion and related phenomena.
8. What is the dimension of viscosity?
The dimension of viscosity is [ML^-1T^-1]. This can be understood from the definition of viscosity as the ratio of shear stress to shear rate. Understanding this dimension is crucial in fluid dynamics and helps in analyzing fluid flow problems.
9. How do dimensions help in understanding the concept of physical universality?
Dimensions relate to physical universality by:
10. What is the dimension of thermal conductivity?
The dimension of thermal conductivity is [MLT^-3θ^-1], where θ represents temperature. This dimension arises from the definition of thermal conductivity in terms of heat transfer rate, temperature gradient, and area. Understanding this dimension is crucial in heat transfer problems.
11. How do dimensions help in understanding the relationships between different physical quantities?
Dimensions help understand relationships between physical quantities by:
12. How do dimensions relate to the concept of physical intuition in problem-solving?
Dimensions enhance physical intuition in problem-solving by:
13. How do dimensions help in understanding the limitations of physical models?
Dimensions help understand the limitations of physical models by:
14. How do dimensions relate to the concept of physical analogies?
Dimensions are fundamental to physical analogies by:
15. How do you determine the dimensions of a derived quantity?
To determine the dimensions of a derived quantity, analyze its relationship with fundamental quantities. For example, for acceleration (a = v/t), divide the dimensions of velocity [LT^-1] by time [T], resulting in [LT^-2]. This process involves multiplying or dividing the dimensions of the constituent quantities based on their mathematical relationship.
16. How can dimensional analysis help in problem-solving?
Dimensional analysis is a powerful problem-solving tool in physics. It can be used to:
17. What is the principle of dimensional homogeneity?
The principle of dimensional homogeneity states that all terms in a physical equation must have the same dimensions. This principle is a fundamental check for the validity of physical equations. If an equation violates this principle, it is incorrect or incomplete.
18. Can dimensions have fractional or negative powers?
Yes, dimensions can have fractional or negative powers. For example, the dimension of energy density is [ML^-1T^-2], where length (L) has a negative power. Fractional powers can occur in more complex derived quantities. These powers arise from the mathematical relationships between physical quantities.
19. What is a dimensionless quantity?
A dimensionless quantity is a physical quantity that has no associated physical dimension. It is a pure number without units. Examples include strain, refractive index, and Reynolds number. These quantities often result from ratios of quantities with the same dimensions.
20. What is the dimension of momentum?
The dimension of momentum is [MLT^-1]. This can be derived from the definition of momentum as mass times velocity. Understanding this dimension is crucial in mechanics and helps in analyzing collisions and conservation laws.
21. What is the dimension of force?
The dimension of force is [MLT^-2]. This can be derived from Newton's Second Law (F = ma), where mass [M] is multiplied by acceleration [LT^-2]. Understanding this dimension helps in analyzing and deriving equations involving force.
22. How do dimensions help in understanding the concept of physical units?
Dimensions help understand physical units by:
23. How do dimensions relate to the concept of physical constants in different unit systems?
Dimensions help understand physical constants across unit systems by:
24. How do dimensions help in understanding the concept of physical symmetry?
Dimensions relate to physical symmetry by:
25. What is dimensional analysis, and how is it used?
Dimensional analysis is a method of analyzing physical relationships using their dimensions. It involves:
26. Can you have a physical quantity with no dimensions?
Yes, there are physical quantities with no dimensions, called dimensionless quantities. These are pure numbers without units. Examples include:
27. How do dimensions help in unit conversion?
Dimensions help in unit conversion by providing a framework for consistency. When converting between units, the dimensions must remain the same. This principle allows for the creation of conversion factors. For example, knowing that both meters and feet are units of length [L] allows for a direct conversion between them.
28. What is the significance of dimensional constants?
Dimensional constants are physical constants that have dimensions. They play crucial roles in physics by relating different physical quantities. Examples include the speed of light (c) and the gravitational constant (G). These constants often appear in fundamental equations and can be used to define natural units of measurement.
29. How do dimensions relate to the SI system of units?
The SI system of units is based on seven base units corresponding to the seven base dimensions: meter (length), kilogram (mass), second (time), ampere (electric current), kelvin (temperature), candela (luminous intensity), and mole (amount of substance). All other SI units are derived from these base units, reflecting the relationships between their dimensions.
30. How do dimensions help in identifying errors in equations?
Dimensions help identify errors in equations by:
31. How do dimensions relate to the concept of similarity in physics?
Dimensions are fundamental to the concept of similarity in physics. Two systems are considered similar if they have the same values for all relevant dimensionless parameters. This concept allows for:
32. How do dimensions help in understanding the nature of physical quantities?
Dimensions help understand the nature of physical quantities by:
33. How do dimensions relate to the concept of physical constants?
Dimensions are integral to understanding physical constants. They:
34. What is the significance of dimensionless numbers in physics?
Dimensionless numbers in physics are important because they:
35. How do dimensions relate to physical laws?
Dimensions are fundamental to physical laws. They ensure that:
36. How can dimensions help in deriving new equations?
Dimensions can help derive new equations by:
37. How do dimensions relate to scaling laws in physics?
Dimensions are crucial in developing scaling laws, which describe how physical quantities change with size or scale. By analyzing dimensions, one can determine how different quantities should scale relative to each other. This is particularly useful in fields like fluid dynamics and structural engineering.
38. What is the Buckingham Pi theorem in dimensional analysis?
The Buckingham Pi theorem is a key principle in dimensional analysis. It states that:
39. What is the dimension of pressure?
The dimension of pressure is [ML^-1T^-2]. This can be derived from the definition of pressure as force per unit area. Force has dimensions [MLT^-2], and area has dimensions [L^2]. Dividing force by area gives [MLT^-2]/[L^2] = [ML^-1T^-2].
40. What is the dimension of energy?
The dimension of energy is [ML^2T^-2]. This can be derived from various definitions of energy, such as kinetic energy (1/2 mv^2) or work (force × distance). Understanding this dimension helps in analyzing energy-related equations and conversions.
41. What is the dimension of power?
The dimension of power is [ML^2T^-3]. This can be derived from the definition of power as energy per unit time. Energy has dimensions [ML^2T^-2], and dividing by time [T] gives [ML^2T^-3]. This dimension is useful in analyzing and deriving equations involving power and energy transfer.
42. How do dimensions relate to the concept of physical approximations?
Dimensions help in understanding and making physical approximations by:
43. How do dimensions help in understanding the concept of physical limits?
Dimensions help understand physical limits by:
44. What is the dimension of electric potential?
The dimension of electric potential is [ML^2T^-3I^-1]. This can be derived from the definition of potential as energy per unit charge. Understanding this dimension is crucial in electrostatics and circuit analysis.
45. What is the dimension of electric charge?
The dimension of electric charge is [IT], where I represents electric current and T represents time. This dimension arises from the definition of current as charge flow per unit time. Understanding this dimension is crucial in electromagnetism and helps in analyzing electrical equations.
46. What is the dimension of magnetic field strength?
The dimension of magnetic field strength is [MT^-2I^-1]. This can be derived from the force on a moving charge in a magnetic field. Understanding this dimension is crucial in electromagnetism and helps in analyzing magnetic phenomena.
47. What is the dimension of surface tension?
The dimension of surface tension is [MT^-2]. This can be understood as force per unit length, reflecting the energy required to increase the surface area of a liquid. This dimension is crucial in understanding phenomena like capillary action and droplet formation.
48. What is the dimension of electric field strength?
The dimension of electric field strength is [MLT^-3I^-1]. This can be derived from the definition of electric field as force per unit charge. Understanding this dimension is crucial in electrostatics and helps in analyzing electric phenomena.
49. What is the dimension of specific heat capacity?
The dimension of specific heat capacity is [L^2T^-2θ^-1]. This arises from the definition of specific heat capacity as the energy required to raise the temperature of a unit mass by one degree. Understanding this dimension is crucial in thermodynamics and heat transfer problems.
50. What is the dimension of magnetic flux?
The dimension of magnetic flux is [ML^2T^-2I^-1]. This can be understood from the definition of flux as the product of magnetic field strength and area. Understanding this dimension is crucial in electromagnetism and helps in analyzing phenomena like electromagnetic induction.

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