Which of the following is not a dimensionless quantity ?

Relative magnetic permeability (µ_r) It is the ratio of permeability of medium to the permeability of free space i.e., ${\mathrm{\mu }}_{\mathrm{r}}=\frac{{\mathrm{\mu }}_{\mathrm{m}}}{{\mathrm{\mu }}_0}$

As, it is a ratio of permeabilities. So, it is unitless and dimensionless quantity.

Power factor $\left(\mathrm{cos\varphi }\right)$ 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 (µ₀) It is the ratio of magnetising field induction (B) to magnetising field intensity (H)

$\begin{array}{l}\mathrm{i}.\mathrm{e}. {\mathrm{\mu }}_0=\frac{\mathrm{B}}{\mathrm{H}}\\ \Rightarrow \left[{\mathrm{\mu }}_0\right]=\frac{\left[\mathrm{B}\right]}{\left[\mathrm{H}\right]} ....\left(\mathrm{i}\right)\\ \mathrm{Since}, \mathrm{H}=\mathrm{nI}\end{array}$

where, n = number of turns per unit length and I = current

$\begin{array}{l}\therefore \left[\mathrm{H}\right]=\left[\mathrm{n}\right][/]\\ =\left[{\mathrm{L}}^{-1}\right]\left[\mathrm{A}\right]=\left[{\mathrm{AL}}^{-1}\right]\end{array}$

We know that, force on a current carrying conductor in magnetic field B is given as

        $\mathrm{F}=\mathrm{IBI}$

where, l = length of the conductor,

             B = magnetic field

and      I = current.

$\begin{array}{cl}\therefore & \mathrm{B}=\frac{\mathrm{F}}{\mathrm{II}}\\ \Rightarrow & \left[\mathrm{B}\right]=\frac{\left[\mathrm{F}\right]}{\left[\mathrm{I}\right][/]}=\frac{\left[{\mathrm{MLT}}^{-2}\right]}{\left[\mathrm{A}\right]\left[\mathrm{L}\right]}=\left[{\mathrm{MT}}^{-2}{\mathrm{A}}^{-1}\right]\end{array}$

$\therefore$ From Eq. (i), we have

      $\left[{\mathrm{\mu }}_0\right]=\frac{\left[{\mathrm{MT}}^{-2}{\mathrm{A}}^{-1}\right]}{\left[{\mathrm{AL}}^{-1}\right]}=\left[{\mathrm{MLT}}^{-2}{\mathrm{A}}^{-2}\right]$

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.

$\text{ Quality factor }=\frac{\text{ Energy stored }}{\text{ Energy dissipated per cycle }}$

As, it is ratio of energies so, it will be unitless and dimensionless quantity.