Dimensional analysis is the practice of checking relations between physical quantities by identifying the dimensions of the physical quantities.
The expression showing the powers to which the fundamental units are to be raised to obtain one unit of a derived quantity is called the dimensional formula of that quantity.
If Q is the unit of a derived quantity represented by Q = MaLbTc, then MaLbTc,is called dimensional formula and the exponents a, b and, c are called the dimensions.
Dimensional formula for Resistance
The dimensional formula of resistance is given by,
M1 L2 T-3 I-2
- M = Mass
- I = Current
- L = Length
- T = Time
Resistance (R) = Voltage × Current-1 ————–(i)
Since, voltage (V) = Electric Field × Distance
= [Force × Charge-1] × Distance
The dimensional formula of charge = current × time = I1 T1
The dimensional formula of voltage = [Force × Charge-1] × Distance
= [M1 L1 T-2] × [I1 T1]-1 × [L1] = [M1 L2 T-3 I-1] ———–(ii)
On substituting equation (ii) in equation (i) we get,
Resistance (R) = Voltage × Current-1
R = [M1 L2 T-3 I-1] × [I]-1 = [M1 L2 T-3 I-2]
Therefore, resistance is dimensionally represented as M L2 T-3 I-2.
Dimensional formula for Inductance
For inductance, the defining equation is,
But ϕ has units [(magnetic field)*(length)]2
Magnetic field from Lorentz force law has units, (Force)(velocity)-1(charge) -1
Therefore, dimensions of magnetic field,[B]=MLT−2 / LT−1AT
[B]=MLT−2 / LA
Therefore dimensions of magnetic flux,[ϕ]=[B]L2
Hence, the dimensions of inductance,[L]=[ϕ][I][L]=MT−2L2A−2
Therefore, inductance is dimensionally represented as MT−2L2A−2
Dimensional formula of
(i) Resistance is M L2 T-3 I-2
(ii) Inductance is MT−2L2A−2