Animated Solution for Physics - Physics and Measurement: Which of the following is not a dimensionless quantity ?

The quest for dimensions is one of the most fundamental and revealing exercises in physics. It forces us to strip away the numerical values and look at the pure, underlying physical nature of a quantity. In this problem, we are presented with four distinct physical quantities and asked to identify the "odd one out"—the one that actually possesses physical dimensions.

Let's embark on this journey by analyzing each option, one by one.

We begin with Relative Magnetic Permeability (${\mu }_r$). By its very definition, relative permeability is a comparative measure. It tells us how permeable a specific medium is compared to the absolute vacuum of free space. Mathematically, it is expressed as:

Because we are dividing a permeability by another permeability, their units perfectly cancel each other out. The result is a pure, dimensionless number.

Next, let's look at the Power Factor. In the realm of alternating current (AC) circuits, the power factor is a crucial metric that describes how effectively electrical power is being converted into useful work output. It is defined as the cosine of the phase angle ($\varphi$) between the voltage and the current:

Any trigonometric function, whether it is sine, cosine, or tangent, fundamentally represents a ratio of two side lengths in a right-angled triangle. A length divided by a length leaves no dimensions behind. Thus, the power factor is entirely dimensionless.

Now, let's jump to the Quality Factor ($Q$). In resonant circuits, the Q-factor measures the "sharpness" or efficiency of the resonance. It is defined as $2\pi$ times the ratio of the maximum energy stored in the circuit to the energy dissipated per cycle:

Here again, we are dividing Joules by Joules. The dimensions of energy in the numerator and denominator annihilate each other, confirming that the Quality Factor is also a dimensionless quantity.

Permeability of Free Space

By elimination, we know our answer must be the Permeability of Free Space (${\mu }_0$). But in physics, we don't just guess; we prove. Let's derive its dimensions from scratch.

We start with the fundamental relationship between the magnetic induction ($B$) and the magnetic intensity ($H$):

To find the dimensions of ${\mu }_0$, we first need the dimensions of $H$. For a long solenoid, the magnetic intensity is given by the product of the turn density and the current:

Since $n$ is the number of turns per unit length, its dimension is $[L^{-1}]$. Current $I$ has the dimension $[A]$. Therefore, the dimension of $H$ is:

Next, we need the dimensions of the magnetic field $B$. We can extract this from the magnetic force experienced by a current-carrying wire:

Substituting the known dimensions for force ($[ML{T}^{-2}]$), current ($[A]$), and length ($[L]$), we get:

We now have all the pieces of the puzzle. Let's substitute the dimensions of $B$ and $H$ back into our equation for ${\mu }_0$:

Look at that result! It is packed with fundamental dimensions: mass, length, time, and current. It is unequivocally not dimensionless.

Therefore, the permeability of free space (${\mu }_0$) is the correct answer. This exercise beautifully demonstrates how dimensional analysis can be used to verify the physical nature of any constant or variable in the universe.