# Dimensions & Dimensional formulae.

Dimensions: Dimensions of a physical quantity are,the powers to which the fundamental units  are raised to get one unit of the physical quantity.

The fundamental quantities are expressed with following symbols while writing dimensional formulas of derived physical quantities.

Mass →[M] ; Length→[L]; Time→[T]; Electric current →[I] ; Thermodynamic temperature →[K] ;Intensity of light →[cd] ; Quantity of matter →[mol] .

Dimensional Formula :Dimensional formula of a derived physical quantity  is the “expression showing powers to which different fundamental units are raised”.

Ex : Dimensional formula of Force  F →[$M^1L^1T^{-2}$]

Dimensional equation:When the dimensional formula of  a physical quantity is expressed in the form of  an equation by writing the physical quantity on the left hand side and the dimensional formula on the right hand side,then the resultant equation is called Dimensional equation.

Ex: Dimensional equation of Energy is E = [$M^1L^2T^{-2}$] .

Question : How can you derive Dimensional formula of a derived physical quantity.

Ans : We can derive dimensional formula of any derived physical quantity in two ways

i)Using the formula of the physical quantity : Ex: let us derive dimensional formula of Force .

Force F→ma ; substitute the dimensional formula of mass m →[M] ; acceleration →[$LT^{-2}$]

we get F → [M][$L T^{-2}$]; F →[$M^1L^1T^{-2}$] .

ii) Using the units of the derived physical quantity. Ex: let us derive the dimensional formula of momentum.

Unit of Momentum ( p ) → [$kg-m sec^{-1}$] ;

kg is unit of mass → [M] ; is unit of length → [L] ; sec is the unit of time →[T]

Substitute these dimensional formulas in above equation we get p →[$M^1L^1T^{-1}$].

Quantities having no units, can not possess dimensions: Trigonometric ratios, logarithmic functions, exponential functions, coefficient of friction, strain, poisson’s ratio, specific gravity, refractive index, Relative permittivity, Relative permeability. All these quantities neither possess units nor dimensional formulas.

Quantities having units, but no dimensions : Plane angle,angular displacement, solid angle.These  physical quantities possess units but they does not possess dimensional formulas.

Quantities having both units & dimensions : The following quantities  are examples of such quantities.

Area, Volume,Density, Speed, Velocity, Acceleration, Force, Energy etc.

Physical Constants : These are two types

i) Dimension less constants (value of these constants will be same in all systems of units): Numbers, pi, exponential functions are dimension less constants.

ii)Dimensional constants(value of these constants will be different in different systems of units): Universal gravitational constant (G),plank’s constant (h), Boltzmann’s constant (k), Universal gas constant (R), Permittivity of free space($\in_0$) , Permeability of free space ($\mu_0$),Velocity of light (c).

Principle of Homogeneity of dimensions: The term on both sides of a dimensional equation should have same dimensions.This is called principle of Homogeneity of dimensions. (or) Every term on both sides of a dimensional equation should have same dimensions.This is called principle of homogeneity of dimensions.

Uses of Dimensional equations : dimensional equations are used i) to convert units from one system to another,

ii)to check the correctness of the dimensional equations iii)to derive the expressions connecting different physical quantities..

Limitations of dimensional method: The limitations of dimensional metthod,s are

i)The value of dimensionless constants can not be calculated using dimensional methods,

ii) We can not analyze the equations containing trigonometrical, exponential and logarithmic functions using method of dimensions.

iii)If a physical quantity is sum or difference of two or more than two physical quantities, such physical quantities can not be derived with dimensional methods,

iv)If  any equation having dimensional constants like, G, R etc can not be derived using dimensional methods,

v)If any equations is involving more than three fundamental quantities in it, such expressions can not be derived using dimensional methods.

## 25 thoughts on “Dimensions & Dimensional formulae.”

1. pls i need 2 know on d topic; co-efficient of a friction as adimensional guantity

2. i need to know the dimention formula of electric permivity

3. Jalmeen

I need to know the dimension formula of the following:-

1. Absolute Electric Permittivity
2. Dielectric constant
3. Resistance
4. Capacitance
5. Conductance
6. Resistivity
7. Conductivity
8. Inductance
9. Impedence
10.Magnetic Permeability
11.Potential
12.Estiffience Constant

4. nikita kale

pls.. tell me how do we find dimension of permittivity of vaccum..??

5. very nice mind blowing!!!!!!!!!!

6. gyaunnrraje

As desired by you the dimensional formula of permittivity is added to table

7. gyaunnrraje

Table is updated.Please go through it and let me know if any of the physical quantities are missing in the table.

8. gyaunnrraje

Please go through the table , now it is updated

9. pankaj

what is dimensional formula of electric current & charge

10. lt. Abdullah

long live sirrr.thank u very much..

11. lt. Abdullah

what is the dimension formula of permeability

12. kawale parijat satish

find the dimensions of permittivity of vaccum

13. gyaunnrraje

The D.F of permittivity of free space(vacuum) will be same as the permittivity of any other medium.The D.F of
permittivity is given in the table.Relative permittivity is a Dimensional less quantity.

14. vishal

ples tel me physics for deploma 1st sameshter imporetant sample paper imgetly

15. Lee

What are the dimensions of the physical quantities in physics?

16. zahoor ahmed shah

fantastic effort sir. God bless you

17. its mind blowing plz tell me the dimension of Q=I*t

18. Sandeep Mand

Physics

19. Sandeep Mand

Physics dimensions

20. Sandeep Mand

plz tell me physics for diploma E.C.E. 1st semester important sample paper imgently

21. Sandeep Mand

plz tell me physics for diploma E.C.E. 1st semester important sample paper .imigently. . .

22. Venkat

Which is the physical quantity that has unit of measurement but has no dimensional formula

23. beauty

What’s the dimensional formula of permittivity in medium

24. Moyo

This is a good work done for the benefit of us, scientist to become

25. james

This helped me much!! Congrats