Animated Solution for Physics - Physics and Measurement: In SI units, the dimensions of $\sqrt{\frac{{\epsilon }_0}{{\mu }_0}}$ is

The Elegant Path

Have you ever looked at a physics problem and thought, "There has to be a better way?" This question is the perfect example of that exact feeling. We are asked to find the dimensional formula for a rather intimidating expression: $\sqrt{\frac{{\epsilon }_0}{{\mu }_0}}$.

If we take the brute force approach, we would first calculate the dimensions of permittivity (${\epsilon }_0$), then calculate the dimensions of permeability (${\mu }_0$), and finally divide them and take the square root. While this method is completely correct and will eventually lead you to the right answer, it is a very long and tedious route.

But we are preparing for competitive exams like JEE, where time is our most valuable asset. We cannot afford to spend five minutes on a question that can be solved in one. We need an elegant path. We need a shortcut that bypasses the heavy lifting.

To find our shortcut, we need to dig into our memory of electromagnetic waves. Enter James Clerk Maxwell. His brilliant synthesis of electromagnetism gave us one of the most beautiful equations in physics, connecting the speed of light with the fundamental properties of the vacuum.

Do you remember the famous relation? The speed of light, denoted by $c$, is equal to:

This single equation is a lifesaver. We can use this to transform our complicated expression into something much simpler. Let us see how we can introduce the speed of light into our given expression.

We have the square root of ${\epsilon }_0$ over ${\mu }_0$. What if we multiply the numerator and the denominator inside the square root by ${\epsilon }_0$? Let's do that:

This gives us ${\epsilon }_0^2$ in the numerator. Now, we can easily take ${\epsilon }_0^2$ out of the square root, leaving us with:

And look at that! The second part is exactly the speed of light, $c$. So, our entire expression simplifies beautifully to just:

Now our task is incredibly simple. Instead of dealing with ${\mu }_0$, we just need the dimensions of ${\epsilon }_0$ and the speed of light $c$.

To find the dimensions of ${\epsilon }_0$, we don't need to memorize it. Memorizing complex dimensional formulas is a recipe for disaster under exam pressure. Let us just recall Coulomb's law from electrostatics. The electrostatic force $F$ between two charges is given by:

By simply rearranging this formula, we can isolate ${\epsilon }_0$:

Let us substitute the basic dimensional formulas into our rearranged equation. The dimension of charge $q$ is current times time, which is $[AT]$. So we have $[A^2{T}^2]$ in the numerator. In the denominator, the dimension of force is $[ML{T}^{-2}]$, and the dimension of distance squared is $[L^2]$.

Now, let us carefully combine these terms. Bringing everything to the numerator, we get the dimensional formula for ${\epsilon }_0$:

Take a moment to verify this calculation. It is a very common derivation, and being able to do it quickly will save you countless times.

We are at the final step. We need to multiply the dimensions of ${\epsilon }_0$ with the dimensions of the speed of light. The speed of light is just a velocity, so its dimensions are simply $[L{T}^{-1}]$.

Let us multiply our previous result by $[L{T}^{-1}]$:

Now, we just combine the like terms. - Combining the $L$ terms: $L^{-3}$ and $L^1$ gives us $L^{-2}$. - Combining the $T$ terms: $T^4$ and $T^{-1}$ gives us $T^3$.

And there we have it! The final dimensional formula is:

This matches perfectly with option (d). A complex problem dismantled by a simple, elegant trick. Always look for these hidden connections in physics; they are the key to mastering the subject!