Checking dimensions in equations

Short post this week.

I want to talk about a simple concept that is extremely useful in physics and other fields of applied science. Especially those fields working with equations to calculate anything that has a physical meaning to it, like a speed, a voltage, a weight..

As a physics student, our teachers would usually spend a few minutes talking about how it is extremely important to check dimensions. Back then, I was listening and not really thinking how that would help me concretely to find the answer to the problem they were giving us. I would always be puzzled by why they were talking about this, and then never, ever, using it themselves in class or during the exercises. It seemed like a gimmick that I could not understand.

As a physics tutor for bachelor students, I made sure to always explain why checking the dimensions is the most important habit to have. I even thought to myself: Why didn’t our teacher spent more time on that? It’s so useful!

I try to make posts related to what I live through. This week, I had to review some technical text and low and behold, there was an equation. Simple equation. To be honest, this is an equation I don’t know by heart, it’s an equation I always derive from other base equations. It’s a habit I have to avoid remembering too many redundant things that I can simply find again (prove again) in less than a minute. That way I found out I make less mistakes.

Anyway, that equation is used to find the optical path length for a given phase difference. The equation I saw was this one:

OPL = \dfrac{2\pi \times \Delta\phi}{\lambda}

It’s wrong. And I could immediately see that in 5 seconds. How? by checking the dimensions of the equation.

First things to know are the dimensions of the components of the equation. OPL is a distance, we can note that [m]. 2\pi is representing an angle in this case specifically, we can note that [rad]. \lambda is the wavelength, it is a distance, we can also note that [m]. \Delta \phi is the phase, it has a dimension of an angle, we note that [rad]…

Let’s replace the components of the equation by the dimensions…

[m] \neq \dfrac{[rad]\times[rad]}{[m]}

As you can notice, the right side of the equation has a strange dimension: angle squared divided by a length, while the left side of the equation is a simple distance… Not equal at all, the equation is false. Do I need to know the details of the equation to know it is false? No. I can tell this equation is false just by this simple trick that cost me 5-10 seconds depending on the complexity of the equation…

Now, this technique is helpful to know if an equation is false, but it certainly doesn’t tell you if the equation is right…


OPL = \dfrac{2\pi \times \lambda}{\Delta\phi}

Make the dimension analysis:

[m] = \dfrac{[rad]\times[m]}{[rad]}

You divide angles by angles, end up with a ratio and the only dimensions left is a distance in both sides of the equation. Yet, the equation is false.

So checking the dimensions is not a proof by itself that an equation is right. It is only a proof that an equation is false or that it could be right…. Nuance!

For the curious, the real, correct equation is the following:

OPL = \dfrac{\Delta\phi\times\lambda}{2\pi}

The dimension analysis would give the same result as the previous equation, ending in a length in both sides of the equation.

Tips and tricks: how to remember this equation? This equation is basically a ratio between phase and length. The main idea is that when your wave (light in this instance) travels a given distance of 1 wavelength, the phase will have turned 360° making a full circle. Because we usually work in radian we use 2\pi instead.

What happens when the wave travels a given distance, here OPL, then the phase will have turn a given amount \Delta \phi. Because this is a simple proportionality relation between OPL and \Delta \phi and we know the proportion for a OPL of 1 wavelength, we can easily find the relation:

\dfrac{OPL}{\Delta\phi} = \dfrac{\lambda}{2\pi}

which is the equation I recreate every time I need to use it instead of remembering by heart.