Animated Solution for Physics - Physics and Measurement: Dimensions of $\frac{1}{{\mu }_0{\epsilon }_0}$, where symbols have their usual meaning, are

When faced with a dimensional analysis problem involving a complex combination of physical constants, the brute-force approach is often the most painful. Imagine trying to find the dimensions of $\frac{1}{{\mu }_0{\epsilon }_0}$ by first deriving the dimensions of the magnetic permeability of free space (${\mu }_0$) and then the electric permittivity of free space (${\epsilon }_0$). You would have to recall formulas like Coulomb's Law and the Biot-Savart Law, extract the dimensions, multiply them together, and then take the reciprocal. It is a recipe for a silly mistake!

Physics is beautiful because it is interconnected. Instead of treating ${\mu }_0$ and ${\epsilon }_0$ as isolated constants, we should ask ourselves: Where have I seen these two constants together before?

The answer lies in the heart of electromagnetism. James Clerk Maxwell discovered that light itself is an electromagnetic wave, and its speed in a vacuum, denoted by $c$, is dictated entirely by these two properties of space. The legendary formula is:

This single equation turns a tedious dimensional analysis problem into a trivial one. We need the dimensions of $\frac{1}{{\mu }_0{\epsilon }_0}$. To get this exact expression, we simply square both sides of Maxwell's equation:

Now, the problem is reduced to finding the dimensions of $c^2$. Since $c$ is the speed of light, it is fundamentally just a velocity. The dimensions of any velocity are distance divided by time:

Substituting this into our squared equation, we get:

And there we have it! The dimensions are $[L^2{T}^{-2}]$. This problem is a perfect reminder that in physics, memorizing the fundamental relationships between quantities is far more powerful than memorizing their individual dimensional formulas.