Units and Measurements Part 1

Relative magnetic permeability $(\mu_{r})$ It is the ratio of permeability of medium to the permeability of free space i.e., $\mu_{r}=\frac{\mu_{m}}{\mu_{0}}$ As, it is a ratio of permeabilities. So, it is unitless and dimensionless quantity. Power factor $(\cos \phi)$ It is cosine of phase difference between alternating current and alternating voltage. Hence, it has only a numerical value, so it is also a unitless and dimensionless quantity. Permeability of free space $(\mu_{0})$ It is the ratio of magnetising field induction (B) to magnetising field intensity (H) i.e. $\mu_{0}=\frac{B}{H}$ $\Rightarrow [\mu_{0}]=\frac{[B]}{[H]}$ ...(i) Since, $H=nl$ where, n = number of turns per unit length and I = current $\therefore [H]=[n][I]$ $=[L^{-1}][A]=[AL^{-1}]$ We know that, force on a current carrying conductor in magnetic field B is given as F = IBl where, l = length of the conductor, B = magnetic field and I = current. $\therefore B=\frac{F}{II}$ $\Rightarrow [B]=\frac{[F]}{[I][I]}=\frac{[MLT^{-2}]}{[A][L]}=[MT^{-2} A^{-1}]$ ∴From Eq. (i), we have $[\mu_{0}]=\frac{[MT^{-2} A^{-1}]}{[AL^{-1}]}=[MLT^{-2} A^{-2}]$ Hence, permeability of free space is not a dimensionless quantity. Quality factor It is the ratio of energy stored to the energy dissipated per cycle. Quality factor $\frac{\text{Energy stored}}{\text{Energy dissipated per cycle}}$ As, it is ratio of energies so, it will be unitless and dimensionless quantity.