Principle of homogeneity of dimensions
It states that in a correct equation, the dimensions of each term added or subtracted must be same. Every correct equation must have same dimensions on both sides of the equation.
Conversion of units
The numerical value of a physical quantity in a system of units can be changed to another system of units using the equation n[u] = constant, i.e., n_{1}[u_{1}] = n_{2}[u_{2}], where n is the numerical value and u is the unit.
By knowing the conversion factors for the base quantities and dimensional formula of the derived quantity, one can convert the numerical value of a physical quantity from one system of units to other system of units.
where the dimensional formula of the physical quantity is [M^{a}L^{b}T^{c}].
To find a relation among the physical quantities
If one knows the quantities on which a particular physical quantity depends and guesses that this dependence is of product type, method of dimensions are helpful in deducing their relation.
Suppose we want to find the relation between force, mass, and acceleration. Let force depends on mass and acceleration as follows:
F = Km^{b}a^{c}
where K is the dimensionless constant and b and c are powers of mass and acceleration.
According to the principle of homogeneity,
[F] = [K] [m]^{b} [a]^{c}
â‡’ [MLT^{â€“2}] = [M^{0}L^{0}T^{0}] [M]^{b} [LT^{â€“2}]^{c}
â‡’ [MLT^{â€“2}] = M^{b}L^{c} T^{â€“2c}
Equating the dimension on both sides, we get
1 = b, 1 = c, â€“2c = â€“2.
â‡’ b = 1 and c = 1.
Now putting the values of b and c in our required equation, we will get a mathematical equation F = Kma. The value of K can be found experimentally.
Table Quantities Having Same Dimensions
Dimension

Quantity

[M^{0}L^{0}T^{â€“1}]

Frequency, angular frequency, angular velocity, velocity gradient and decay constant

[M^{1}L^{2}T^{â€“2}]

Work, internal energy, potential energy, kinetic energy, torque, moment of force

[M^{1}L^{â€“1}T^{â€“2}]

Pressure, stress, Youngâ€™s modulus, bulk modulus, modulus of rigidity, energy density

[M^{1}L^{1}T^{â€“1}]

Momentum, impulse

[M^{0}L^{1}T^{â€“2}]

Acceleration due to gravity, gravitational field intensity

[M^{1}L^{1}T^{â€“2}]

Thrust, force, weight, energy gradient

[M^{1}L^{2}T^{â€“1}]

Angular momentum, Planckâ€™s constant

[M^{1}L^{0}T^{â€“2}]

Surface tension, surface energy (energy per unit area)

[M^{0}L^{0}T^{0}]

Strain, refractive index, relative density, angle, solid angle, distance gradient, relative permittivity (dielectric constant), relative permeability.

[M^{0}L^{2}T^{â€“2}]

Latent heat, gravitational potential

[ML^{2}T^{â€“2}Î¸^{â€“1}]

Thermal capacity, gas constant, Boltzmann constant, entropy

[M^{0}L^{0}T^{1}]

where l = length, g = acceleration due to gravity, m= mass, k = spring constant,R = radius of Earth

[M^{0}L^{0}T^{1}]

L/R, , RC where L = inductance, R = resistance, C= capacitance

[ML^{2}T^{â€“2}]

where I = current, t = time, q= charge,
L = inductance, C = capacitance, R = resistance
