JEE PYQ: Units & Measurements - Question ID 265f518d24bb (JEE Main 2021)

In the chapter Units & Measurements, a dimensionless quantity is defined as a physical quantity that has no physical dimensions associated with it. In other words, its dimensional formula reduces to $[M^0 L^0 T^0 A^0]$, meaning it is a pure number without any units like meters, kilograms, seconds, or amperes.

To determine whether a quantity is dimensionless, we analyze its dimensional formula. The dimensional formula expresses the quantity in terms of fundamental dimensions: mass ($M$), length ($L$), time ($T$), electric current ($A$), temperature ($K$), etc.

Key formulas and concepts involved:

• Relative magnetic permeability ($\mu_r$): $\mu_r = \frac{\mu}{\mu_0}$, where $\mu$ is the permeability of a material and $\mu_0$ is the permeability of free space. Since both have the same units, $\mu_r$ is a ratio of two quantities with identical dimensions → dimensionless.
• Power factor: Defined as $\cos \phi$, where $\phi$ is the phase angle between voltage and current in an AC circuit. Since $\phi$ is an angle, it is dimensionless, and so is its cosine → dimensionless.
• Permeability of free space ($\mu_0$): From Maxwell’s equations and the Biot-Savart law, $\mu_0$ relates magnetic field to current. Its SI unit is henry per meter ($\text{H/m}$), and its dimensional formula is $[M L T^{-2} A^{-2}]$ → not dimensionless.
• Quality factor (Q): Defined as $Q = 2\pi \times \frac{\text{Energy stored}}{\text{Energy dissipated per cycle}}$. Both energy terms have the same units, so $Q$ is a ratio → dimensionless.

Step 1: Analyze Option A – Relative magnetic permeability ($\mu_r$)

$\mu_r = \frac{\mu}{\mu_0}$
Since $\mu$ and $\mu_0$ have the same units (henry per meter), their ratio is a pure number.
Dimensional formula: $\frac{[M L T^{-2} A^{-2}]}{[M L T^{-2} A^{-2}]} = [M^0 L^0 T^0 A^0]$
→ $\mu_r$ is dimensionless.

Step 2: Analyze Option B – Power factor

Power factor = $\cos \phi$
$\phi$ is the phase angle, which is dimensionless (measured in radians or degrees, but radians are dimensionless).
Thus, $\cos \phi$ is also dimensionless.
→ Power factor is dimensionless.

Step 3: Analyze Option C – Permeability of free space ($\mu_0$)

From the Biot-Savart law: $dB = \frac{\mu_0}{4\pi} \frac{I \, dl \sin \theta}{r^2}$
Rearranged: $\mu_0 = \frac{4\pi r^2 dB}{I \, dl \sin \theta}$
Units: - $dB$: tesla ($\text{T}$) = $\text{kg s}^{-2} \text{A}^{-1}$ - $r^2$: $\text{m}^2$ - $I$: ampere ($\text{A}$) - $dl$: meter ($\text{m}$)
So, $\mu_0$ has units: $\frac{\text{m}^2 \cdot \text{kg s}^{-2} \text{A}^{-1}}{\text{A} \cdot \text{m}} = \text{kg m s}^{-2} \text{A}^{-2} = \text{H/m}$
Dimensional formula: $[M L T^{-2} A^{-2}]$
→ $\mu_0$ is not dimensionless.

Step 4: Analyze Option D – Quality factor (Q)

$Q = 2\pi \times \frac{\text{Energy stored}}{\text{Energy dissipated per cycle}}$
Both energy terms have units of joule ($\text{J} = \text{kg m}^2 \text{s}^{-2}$), so their ratio is dimensionless.
→ $Q$ is dimensionless.

Conclusion:

Only option C ($\mu_0$) has physical dimensions. Hence, C is not a dimensionless quantity.

Students often confuse relative permeability ($\mu_r$) with absolute permeability ($\mu$ or $\mu_0$). While $\mu_r$ is dimensionless, $\mu_0$ has units and dimensions. Another common mistake is assuming that all constants in physics are dimensionless — this is not true. Always derive the dimensional formula from fundamental laws (e.g., Biot-Savart or Maxwell’s equations) to avoid errors.

Exam Tip: When in doubt, recall that any quantity defined as a ratio of two quantities with the same units is dimensionless. Conversely, fundamental constants like $\mu_0$, $\epsilon_0$, or $G$ usually have dimensions.