Animated Solution for Physics - Physics and Measurement: Which of the following combinations has the dimension of electrical resistance (${ϵ}_0$ is the permittivity of vacuum and ${\mu }_0$ is the permeability of vacuum)?

Dimensional analysis is one of the most powerful tools in a physicist's arsenal. It allows us to uncover hidden relationships between seemingly unrelated physical quantities. In this problem, we are tasked with finding a combination of the permittivity of free space, ${ϵ}_0$, and the permeability of free space, ${\mu }_0$, that yields the dimensions of electrical resistance, $R$.

At first glance, it might seem like magic that these fundamental constants of electromagnetism can combine to form resistance. But as we will see, the math reveals a beautiful underlying structure. Let's break this down step-by-step.

Recalling Dimensions

Before we can build our combination, we need to know the dimensions of our building blocks. If you don't have these memorized, don't panic! You can always derive them from fundamental formulas.

For electrical resistance, $R$, we can use Ohm's law and the definition of electrical work:

Substituting the dimensions of work ($[M{L}^2{T}^{-2}]$), current ($[A]$), and time ($[T]$), we get:

Next, for the permittivity of free space, ${ϵ}_0$, we turn to Coulomb's law:

Using the dimensions of charge ($[AT]$), force ($[ML{T}^{-2}]$), and distance ($[L]$), we find:

Finally, for the permeability of free space, ${\mu }_0$, we can use the formula for the magnetic force between two parallel currents:

This gives us the dimensions:

Setting up the Algebra

Now that we have our dimensions, we can set up a general equation. We want to find powers $\alpha$ and $\beta$ such that:

Let's substitute the dimensional formulas we just found into this master equation:

Using the laws of exponents, we can combine the terms on the right side:

Solving for the Powers

For this equation to hold true, the exponents of each fundamental quantity must be equal on both sides. This gives us a system of linear equations. Let's start by comparing the exponents of mass ($M$) and length ($L$):

For Mass ($M$):

For Length ($L$):

We have two equations and two unknowns. Let's subtract the first equation from the second to eliminate $\beta$:

Now, substitute this value of $\alpha$ back into the first equation to find $\beta$:

We have found our powers! Let's plug them back into our original assumption:

This can be rewritten in a much more elegant form:

And there we have it! The combination $\sqrt{\frac{{\mu }_0}{{ϵ}_0}}$ has the exact dimensions of electrical resistance.

This isn't just a mathematical trick. In physics, the quantity $\sqrt{\frac{{\mu }_0}{{ϵ}_0}}$ is known as the impedance of free space (or vacuum impedance), denoted by $Z_0$. It relates the magnitudes of the electric and magnetic fields of electromagnetic radiation traveling through a vacuum. Its value is exactly $120\pi$ ohms, which is approximately $377\Omega$. So, it makes perfect physical sense that this combination has the dimensions of resistance!