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

In physics, that language is dimensional analysis. Every physical quantity, no matter how complex, is built from a few fundamental building blocks: Mass ($M$), Length ($L$), Time ($T$), and Electric Current ($A$).

Today, we are going to decode the dimensional formula for a very famous constant: the permittivity of free space, denoted by ${\epsilon }_0$. This constant tells us how much resistance a vacuum offers to the formation of an electric field.

Coulomb's Law To find the dimensions of any constant, we first need a reliable equation where it makes an appearance. For ${\epsilon }_0$, the most classic and straightforward equation is Coulomb's Law, which describes the electrostatic force between two point charges.

The formula is:

Here, $F$ is the force, $q_1$ and $q_2$ are the charges, and $R$ is the distance between them.

So, let's rearrange our master equation to isolate ${\epsilon }_0$ on one side:

Now, we are ready to substitute the fundamental dimensions for each variable on the right side.

We know that electric current ($I$) is the rate of flow of charge ($q/t$). Therefore, charge is current times time. Its dimensional formula is $[AT]$. 2. Force ($F$): From Newton's Second Law ($F=ma$), force is mass times acceleration. Its dimensional formula is $[ML{T}^{-2}]$. 3. Distance ($R$): Distance is simply length, so its dimension is $[L]$, and $R^2$ becomes $[L^2]$. 4. Constant ($4\pi$): Pure numbers and mathematical constants are dimensionless. We can safely ignore $4\pi$ in our dimensional analysis.

Now, let's plug these building blocks back into our rearranged equation:

Let's simplify the numerator and the denominator:

Finally, we bring all the terms to the numerator by flipping the signs of their exponents:

And there we have it! The dimensional formula for the permittivity of free space. This matches option (b) perfectly. Dimensional analysis is a fantastic tool not just for finding units, but for checking if your derived equations are physically consistent. Always keep this tool sharp!