Animated Solution for Physics - Physics and Measurement: The quantities $x=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}$, $y=\frac{E}{B}$ and $z=\frac{l}{CR}$ are defined, where $C$ is capacitance, $R$ is resistance, $l$ is length, $E$ is electric field, $B$ is magnetic field, ${\epsilon }_0$ is free space permittivity and ${\mu }_0$ is permeability, respectively. Then,

Imagine you are sitting in the exam hall, the clock is ticking, and you are faced with a monstrous-looking dimensional analysis problem. You see ${\mu }_0$, ${\epsilon }_0$, electric fields, magnetic fields, capacitance, and resistance all jumbled together. The immediate instinct of many students is to roll up their sleeves and start deriving the dimensional formula for each individual term.

But wait! There is a catch here. While the brute-force method of substituting individual dimensions will eventually get you the right answer, it is a massive time sink and a minefield for silly mistakes. This question is not testing your ability to do tedious algebra; it is testing your physical intuition and your ability to recognize standard formulas. Let's break it down elegantly.

The Speed of Light

Let's look at our first quantity, $x=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}$.

Does this expression look familiar? It absolutely should! This is one of the most famous equations in physics, derived from Maxwell's equations. It tells us that the speed of light in a vacuum, $c$, is fundamentally tied to the electric permittivity (${\epsilon }_0$) and magnetic permeability (${\mu }_0$) of free space.

Since $x$ is literally the formula for the speed of light, we don't need to calculate anything. We instantly know its dimension:

The Electromagnetic Wave

Now, let's shift our focus to the second quantity, $y=\frac{E}{B}$.

Here, we have the ratio of the electric field ($E$) to the magnetic field ($B$). Think back to the chapter on Electromagnetic Waves. When an EM wave propagates through space, the magnitudes of its electric and magnetic fields are strictly related by the speed of the wave, $v$. The relation is simply $v=\frac{E}{B}$.

Once again, the physics does the heavy lifting for us. Since $y$ represents the speed of the wave, its dimension must be that of velocity:

The RC Time Constant

Finally, let's tackle the third quantity, $z=\frac{l}{CR}$.

This one looks a bit different. We have a length ($l$) divided by the product of capacitance ($C$) and resistance ($R$). To crack this, we need to remember our Current Electricity concepts. In an RC circuit, the product $CR$ is known as the time constant, often denoted by $\tau$. It literally represents a duration of time!

So, dimensionally, the denominator $[CR]$ is simply $[T]$. Substituting this back into our expression for $z$, we get length divided by time:

Take a step back and look at what we've discovered. Without deriving a single complex dimensional formula, we used our conceptual knowledge to reveal the true nature of these expressions.

We found that:

All three quantities—despite looking completely different and originating from different chapters of physics—share the exact same dimension. They all represent velocity! This is the beauty of physics; seemingly disparate concepts are often deeply interconnected. By trusting your intuition and recognizing standard formulas, you can bypass pages of calculations and arrive at the answer in seconds.