eduFied | edupediaclassesScience |Class 11th| Notes | Physics |Unit- 1. Physical World And Measurements Chapter-1 Physical World
Part-4

Dimensions of physical quantities

Dimension of Physical Quantities:-

The dimensions of a physical zodiac are the powers that have to be mounted on the original units to get the derived units of that physical zodiac.

(The dimensions of a physical quantity are the powers raised on fundamental units to obtain the derived unit of that physical quantity).

for example:-

                Density = mass / volume

 

                              = mass / (length x width x thickness)

 

Drive unit for density     = unit of mass / (unit of length)3

                                       = (unit of mass)1(unit of length)2

 

Obviously: to get the unit of density, we have to raise the power of 1 on the unit of mass and (-3) on the unit of length.
Therefore, the dimensions of density are 1 in mass and (-3) in length.
There is a clear difference between the dimension of a physical sign and its unit.

For example:- the unit of density can be written in different ways like this: gram / cm3, kg / m3, pound / ft.3
Clearly, all these units are different, although each dimension is 1 in mass and (-3) in length.
Therefore, the dimensions of a physical zodiac do not depend on the units expressing the zodiac.
To write the dimensions of the physical quantities in mechanics and thermodynamics, we represent the basic units of mass, length, time and heat with M, L, T and θ respectively.
If the dimensions of a physical zodiac are A in mass, B in length, C in time and D in heat, then the dimensions of that physical zodiac will be written in square bracket as follows: [Ma Lb Tc θd].
This is also called the Dimensional Formula of that physical zodiac.

Dimension of some Physical Quantities:-

Base quantity:-

 

    

Base quantity

Symbol

 

SI base unit

Dimension

 

Length

l

 

metre (m)

L

 

Mass

m

 

kilogram (k)

M

 

Time

t

 

second (s)

T

 

Electric Current

I

 

ampere (A)

I

 

Temperature

T

 

kelvin (K)

Θ

 

Amount of substance

n

 

mole (mol)

N

 

Luminous intensity

Iv, cd

 

candela (cd)

J

 


Derived quantity:-


 

Derived quantity

Symbol

 

SI derived unit

Dimension

 

Absement

A

 

m⋅s

L T

 

Absorbed dose rate

 

 

Gy/s

L2 T−3

 

Acceleration

a

 

m/s2

L T−2

 

Angular acceleration

ωa

 

rad/s2

T−2

 

Angular momentum

L

 

kg⋅m2/s

M L2 T−1

 

Angular velocity

ω

 

rad/s

T−1

 

Area

A

 

m2

L2

 

Area density

ρA

 

kg⋅m−2

M L−2

 

Capacitance

C

 

farad (F = C/V)

M−1 L−2 T4 I2

 

Catalytic activity concentration

 

 

kat⋅m−3

L−3 T−1 N

 

Chemical potential

μ

 

J/mol

M L2 T−2 N−1

 

Crackle

c

 

m/s5

L T−5

 

Current density

 

A/m2

L−2 I

 

Dose equivalent

H

 

sievert (Sv = m2/s2)

L2 T−2

 

Dynamic viscosity

v

 

Pa⋅s

M L−1 T−1

 

Electric charge

Q

 

coulomb (C = A⋅s)

T I

 

Electric charge density

ρQ

 

C/m3

L−3 T I

 

Electric displacement field

D

 

C/m2

L−2 T I

 

Electric field strength

E

 

V/m

M L T−3 I−1

 

Electrical condauctance

G

 

siemens (S = Ω−1)

M−1 L−2 T3 I2

 

Electrical conductivity

σ

 

S/m

M−1 L−3 T3 I2

 

Electric potential

φ

 

volt (V = J/C)

M L2 T−3 I−1

 

Electrical resistance

R

 

ohm (Ω = V/A)

M L2 T−3 I−2

 

Electrical resistivity

ρe

 

ohm-metre (Ω⋅m)

M L3 T−3 I−2

 

Energy

E

 

J

M L2 T−2

 

Energy density

ρE

 

J⋅m−3

M L−1 T−2

 

Entropy

S

 

J/K

M L2 T−2 Θ−1

 

Force

F

 

newton (N = kg⋅m⋅s−2)

M L T−2

 

Frequency

f

 

hertz (Hz = s−1)

T−1

 

Half-life

t1/2

 

s

T

 

Heat

Q

 

joule (J)

M L2 T−2

 

Heat capacity

Cp

 

J/K

M L2 T−2 Θ−1

 

Heat flux density

ϕQ

 

W/m2

M T−3

 

Illuminance

Ev

 

lux (lx = cd⋅sr/m2)

L−2 J

 

Impedance

Z

 

ohm (Ω)

M L2 T−3 I−2

 

Impulse

J

 

newton-second (N⋅s = kg⋅m/s)

M L T−1

 

Inductance

L

 

henry (H)

M L2 T−2 I−2

 

Irradiance

E

 

W/m2

M T−3

 

Intensity

I

 

W/m2

I

 

Jerk

j

 

m/s3

L T−3

 

Jounce (or snap)

s

 

m/s4

L T−4

 

Linear density

ρl

 

kg⋅m−1

M L−1

 

Luminous flux (or luminous power)

F

 

lumen (lm = cd⋅sr)

J

 

Mach number (or mach)

M

 

unitless

1

 

Magnetic field strength

H

 

A/m

L−1 I

 

Magnetic flux

Φ

 

weber (Wb)

M L2 T−2 I−1

 

Magnetic flux density

B

 

tesla (T = Wb/m2)

M T−2 I−1

 

Magnetization

M

 

A/m

L−1 I

 

Mass fraction

x

 

kg/kg

1

 

(Mass) Density (or volume density)

ρ

 

kg/m3

M L−3

 

Mean lifetime

τ

 

s

T

 

Molar concentration

C

 

mol⋅m−3

L−3 N

 

Molar energy

 

 

J/mol

M L2 T−2 N−1

 

Molar entropy

 

 

J/(K⋅mol)

M L2 T−2 Θ−1 N−1

 

Molar heat capacity

c

 

J/(K⋅mol)

M L2 T−2 Θ−1 N−1

 

Moment of inertia

I

 

kg⋅m2

M L2

 

Momentum

p

 

kg⋅m/s

M L T−1

 

Permeability

μs

 

H/m

M L T−2 I−2

 

Permittivity

εs

 

F/m

M−1 L−3 T4 I2

 

Plane angle

θ

 

radian (rad)

1

 

Power

P

 

watt (W)

M L2 T−3

 

Pressure

p

 

pascal (Pa = N/m2)

M L−1 T−2

 

Pop

p

 

m/s6

L T−6

 

(Radioactive) Activity

A

 

becquerel (Bq = Hz)

T−1

 

(Radioactive) Dose

D

 

gray (Gy = m2/s2)

L2 T−2

 

Radiance

L

 

W/(m2⋅sr)

M T−3

 

Radiant intensity

I

 

W/sr

M L2 T−3

 

Reaction rate

r

 

mol/(m3⋅s)

N L−3 T−1

 

Refractive index

n

 

unitless

1

 

Reluctance

{\displaystyle {\mathcal {R}}}

 

H−1

M−1 L−2 T2 I2

 

Solid angle

Ω

 

steradian (sr)

1

 

Specific energy

 

 

J⋅kg−1

L2 T−2

 

Specific heat capacity

c

 

J/(K⋅kg)

L2 T−2 Θ−1

 

Specific volume

v

 

m3⋅kg−1

M−1 L3

 

Spin

S

 

kg⋅m2⋅s−1

M L2 T−1

 

Strain

ε

 

unitless

1

 

Stress

σ

 

Pa

M L−1 T−2

 

Surface tension

γ

 

N/m or J/m2

M T−2

 

Temperature gradient

{\displaystyle \nabla T}

 

K/m

Θ L−1

 

Thermal conductance

 

 

W/K

M L2 T−3 Θ−1

 

Thermal conductivity

λ

 

W/(m⋅K)

M L T−3 Θ−1

 

Thermal resistance

R

 

K/W

M−1 L−2 T3 Θ

 

Thermal resistivity

Rλ

 

K⋅m/W

M−1 L−1 T3 Θ

 

Torque

τ

 

newton-metre (N⋅m)

M L2 T−2

 

Velocity

v

 

m/s

L T−1

 

Volume

V

 

m3

L3

 

Volumetric flow rate

Q

 

m3⋅s−1

L3 T−1

 

Wavelength

λ

 

m

L

 

Wavenumber

k

 

m−1

L−1

 

Wavevector

k

 

m−1

L−1

 

Weight

w

 

newton (N = kg⋅m/s2)

M L T−2

 

Work

W

 

joule (J)

M L2 T−2

 

Young's modulus

E

 

pascal (Pa = N/m2)

M L−1 T−2

 

 

Dimensionless Quantities:-

There are no dimensions of pure numbers and pure proportions, such quantities are called dimensionless quantities, such as - angle, distortion, relative density, π, sinθ, cosθ, tanθ etc.

Uses of Dimensions

The dimensions have the following four uses:

(i) Converting physical quantities from one method of the unit to another

(ii) checking the veracity of an equation

(iii) talking about constants and variables in an equation.

(iv) To establish relations between various physical quantities.