What is called a Dimension?
All physical quantities are expressed in terms of some combination of symbolic representation of fundamental or base quantities. These symbols are called dimension.
EXAMPLE –
Density of a substance is defined as mass per unit volume.
\text {Density} = \left ( \frac { \text {Mass} }{ \text {Volume} } \right )
\text {Dimension of density} = \left [ \frac {M}{{L}^{3 }} \right ] = \left [ M^1 \ L^{ -3 } \ T^0 \right ]
Dimensions are generally expressed in terms of seven fundamental quantities. These seven fundamental quantities are called as the seven dimensions of the world. These are –
- For length = [ L ]
- For mass = [ M ]
- For time, seconds = [ T ]
- For electric current, Ampere = [ I ]
- For thermodynamic temperature, Kelvin = [ K ]
- For luminous intensity, Candela = [ \text {cd} ]
- For amount of substance, mole = [ \text {mol} ]
Out of the above 7 fundamental quantities, 3 quantities are most important and extensively used in dimensional analysis viz. [M] \ [ L ] \ \& \ [T]
Dimensional Analysis
Dimensional analysis is defined as the method of study of a physical phenomenon on the basis of dimensions of involved fundamental quantities in that phenomenon.
Thus, dimensional analysis facilitate the evaluation and formulation of relationship between the involved fundamental quantities in a physical event or phenomenon.
Following are main uses of dimensional analysis –
1. Conversion of System of Units
By use of dimensional analysis, it is possible to convert a physical quantity from one system of units to another. It is based on the fact that the magnitude of the physical quantity remains the same whatever be the system of measurement of units.
2. Checking Correctness of Physical Relations
Dimensional analysis is used to check the correctness of a physical relation.
Principle of Homogeneity of Dimensions
According to this principle, a physical equation will be dimensional correct if the dimensions of all the terms occurring on both sides of the equation are the same.
This principle is based on the fact that, only the physical quantities of the same kind can be added, subtracted or compared. Thus, velocity can be added to velocity but not to force.
3. Deducing Relation among Physical Quantities
If we know the various factors on which a physical quantity depends, then by the use of principle of homogeneity of dimensions, we can derive an expression for that physical quantity.
Dimension of Refractive Index
Refractive Index of a medium is defined as the ratio of the speed of light in vacuum to the speed of light in that medium.
\text {Refractive Index of a medium} = \frac {\text {speed of light in vacuum}}{\text {speed of light in medium}}
\therefore \quad n = \frac {c}{v}
Since, the speed of light in vacuum and the speed of light in medium have the units and dimensions, their ratio is a dimensionless quantity. Therefore, dimension of Refractive Index can be written as follows –
\text {Dimension of Refractive Index} = \left [{M}^{0}{L}^{0}{T}^{0} \right ]
Dimension of Force
As per Newton’s second law of motion, force acting on a body is proportional to the product of mass and acceleration.
P \propto \text {Mass} \times \text {Acceleration}
Therefore, dimension of force will be the product of dimensions of mass and dimension of acceleration.
\text {Dimension of Force} = \left [ {M}^{1} \right ] \times \left [ \frac {{L}^{1}}{{T}^{2}} \right ] = \left [ {M}^{1}{L}^{1}{T}^{-2} \right ]
Dimension of Torque
The turning effect produced by a force is called a moment of a force or torque. Thus, a torque is measured by the product of the force and perpendicular distance of its line of action from the axis of rotation.
\tau = \text {Force} \times \text {Distance}
Therefore, dimension of torque will be written as the product of dimensions of force and dimension of distance –
\text {Dimension of Torque} = \left [ {M}^{1}{L}^{1}{T}^{-2} \right ] \times \left [ {L}^{1} \right ] = \left [ {M}^{1}{L}^{2}{T}^{-2} \right ]
Dimension of Wavelength
Wavelength is defined as the distance covered by a propagating wave in time in which the medium particles complete one cycle of vibration. It may also be defined as the distance between two nearest particles of the medium which are vibrating in the same phase.
Therefore, wavelength has the length attribute and its dimension will be written as –
\text {Dimension of Wavelength} = \left [ {M}^{0}{L}^{1}{T}^{0} \right ]
Dimension of Viscosity
Viscosity is a property of a fluid by virtue of which an internal force of friction comes into play, when the fluid is in motion or tends to be in motion. This frictional force opposes the relative motion between different layers of that fluid. This frictional force is called viscous force. It is found –
- Proportional to the surface area of the fluid layers in contact.
- Proportional to the velocity gradient between the two fluid layers.
Hence, co-efficient of viscosity is defined as –
\eta = \left ( \frac {\text {Tangential viscous force}}{\text {Velocity gradient}} \right ) = \left ( \frac {F / A}{v / x} \right ) = \left ( \frac {F}{A}. \frac {x}{v} \right )
Therefore, dimension of co-efficient of viscosity will be written as –
\text {Dimension of Viscosity} = \left [ \frac {{M}^{1}{L}^{1}{T}^{-2}}{{L}^{2}} \right ] \times \left [ \frac {{L}^{1}}{{L}^{1}{T}^{-1}} \right ] = \left [ {M}^{1}{L}^{-1}{T}^{-1} \right ]
Dimension of Thermal Conductivity
Thermal conductivity is a property of solid which regulates the process of heat transfer in by mode of heat conduction. It is governed by the Fourier’s law of heat conduction.
The coefficient of thermal conductivity of a material is defined as the quantity of heat that flows per unit time through a unit cube of the material, when its opposite faces are kept at a temperature difference of one degree.
K = \left [ \frac{ Q \ L }{ A \left (T_1 - T_2 \right ) t } \right ]
Dimension of heat energy is \quad Q = {M}^{1}{L}^{2}{T}^{-2} .
Therefore, dimension of thermal conductivity will be written as –
\text {Dimension of Thermal Conductivity} = \left [ \frac {\left ({M}^{1}{L}^{2}{T}^{-2} \right ) {L}^{1}}{{L}^{2}{K}^{1}{T}^{1}} \right ] = \left [ {M}^{1}{L}^{1}{T}^{-3}{K}^{-1} \right ]
Dimension of Electrical Resistance
Electrical resistance of a conductor is defined as the ratio of the potential difference across the ends of conductor to the current flowing through that conductor. It has unit of measurement called ohm.
From ohm’s law, we have got the relation between resistance, current and applied voltage as follows –
\text {Resistance ( R )} = \frac { \text {Applied Voltage ( V )}}{ \text {Current Flow ( A )}}
Also work done by electric current is given by, \text {W} = \text {V} \times \text {q} = \text {IR} \times \text {It}
\text {R} = \frac {W}{I^2 t}
Therefore, dimension of electrical resistance can be written as –
\text {Dimension of Resistance} = \left [ \frac {{M}^{1}{L}^{2}{T}^{-2}}{{I}^{2}{T}^{1}} \right ] = \left [ {M}^{1}{L}^{2}{T}^{-3}{I}^{-2} \right ]
Dimension of Resistivity
Specific resistance or resistivity of a material is defined as the electrical resistance offered by that material of unit length and unit cross sectional area. Hence, resistivity is the resistance of a material body of unit volume.
Since, \quad R = \rho \left ( \frac { l }{ A } \right ) \quad \therefore \rho = \left ( \frac { R \ A }{ l } \right )
Hence, dimension of resistivity can be written as –
\text {Dimension of Resistivity} = \left [ \frac {{M}^{1}{L}^{2}{T}^{-3}{I}^{-2} \times {L}^{2}}{{L}^{1}} \right ] = \left [ {M}^{1}{L}^{3}{T}^{-3}{I}^{-2} \right ]
Dimension of Magnetic Field
Magnetic field is defined as the space around a magnet in which its influence of magnetic force of attraction or repulsion is experienced by other small magnets.
This force of attraction or repulsion experienced by an unit north pole strength placed in the magnetic field is called the magnetic field intensity or strength of magnetic field.
Therefore, \quad \text {Intensity of magnetic field (B)} = \frac {\text {Magnetic force on unit North pole (F)}}{\text {Pole strength} (q_m)}
The unit of magnetic pole strength is Ampere-meter or newton / tesla. Hence, its dimension will be \quad \left [{M}^{0}{L}^{1}{T}^{0}{I}^{1} \right ]
Therefore, dimension for the intensity of magnetic field will be written as –
\text {Dimension of Intensity of Magnetic Field} = \left [ \frac {{M}^{1}{L}^{1}{T}^{-2}}{{M}^{0}{L}^{1}{T}^{0}{I}^{1}} \right ] = \left [ {M}^{1}{L}^{0}{T}^{-2}{I}^{-1} \right ]