Animated Solution for Physics - Physics and Measurement: Let $[{\epsilon }_0]$ denote the dimensional formula of the permittivity of the vacuum and $[{\mu }_0]$ that of the permeability of the vacuum. If $M=\text{mass}$, $L=\text{length}$, $T=\text{time}$ and $I=\text{electric current}$.

Have you ever wondered what the vacuum of space is actually made of? In classical electromagnetism, the vacuum isn't just empty nothingness; it has physical properties! Two of the most fundamental properties of the universe are the permittivity of free space (${\epsilon }_0$) and the permeability of free space (${\mu }_0$). These constants dictate how electric and magnetic fields propagate through the void.

In this problem, we are tasked with finding the dimensional formulas for both of these constants. While it might seem like a daunting task to memorize them, we can easily derive them using two of the most famous equations in physics. Let's dive in!

Unlocking Permittivity (${\epsilon }_0$) with Coulomb's Law

To find the dimensions of ${\epsilon }_0$, we need an equation where it naturally appears. The most obvious choice is Coulomb's Law, which describes the electrostatic force between two point charges:

Our goal is to isolate ${\epsilon }_0$. Rearranging the equation, we get:

Now, we perform dimensional analysis. Remember, pure numbers like $4\pi$ are dimensionless, so we can ignore them. We need the dimensions of charge ($q$), force ($F$), and distance ($r$).

A common trap is to assume charge is a fundamental quantity. In the SI system, electric current ($I$) is the fundamental quantity. Since current is the rate of flow of charge ($I=q/t$), we can write charge as $q=It$. Therefore, the dimension of charge is:

The dimensions of force and distance are standard:

Substituting these into our rearranged equation:

Bringing everything to the numerator, we arrive at the dimensional formula for permittivity:

This perfectly matches option (b)!

The Speed of Light Shortcut for Permeability (${\mu }_0$)

Now, we need the dimensions of ${\mu }_0$. We could use the Biot-Savart law or the formula for the magnetic force between two wires. However, there is a much more elegant and faster way.

James Clerk Maxwell discovered a profound connection between electricity, magnetism, and light. He proved that the speed of light in a vacuum ($c$) is completely determined by ${\epsilon }_0$ and ${\mu }_0$:

This equation is a lifesaver for dimensional analysis! Let's square both sides to get rid of the square root:

Rearranging for ${\mu }_0$:

We already know the dimensions of ${\epsilon }_0$. The speed of light ($c$) is simply a velocity, so its dimension is:

Squaring the velocity gives:

Now, let's substitute everything into our equation for ${\mu }_0$:

First, let's simplify the denominator by combining the $L$ and $T$ terms: - For $L$: $-3+2=-1$ - For $T$: $4-2=2$

So the denominator becomes $[{\text{M}}^{-1}{\text{L}}^{-1}{\text{T}}^2{\text{I}}^2]$.

Finally, bringing these terms to the numerator flips the sign of their exponents:

This matches option (c)!

By leveraging Coulomb's Law and Maxwell's beautiful relation for the speed of light, we successfully derived the dimensions of the two constants that govern the vacuum of our universe.

The correct options are (b) and (c).

Always remember: you don't need to memorize every complex dimensional formula. If you know the fundamental equations that connect them, you can derive anything on the spot!