Same equation, different meanings

Question

I went into a physics classroom today and saw this equation written on the board:

$$ E = \frac \sigma \epsilon $$

At first I thought it referred to the electric field $ E $ between 2 parallel plates of charge density $\sigma$ separated by a material of permittivity $\epsilon$. However, I then realised it was actually the definition of the Young's Modulus $E$ for a material that has a strain $\epsilon$ when a stress $\sigma$ is applied to it!

So the same equation has two completely different meanings in two completely different areas of physics, with the symbols defined differently (ignoring symbols to show which variables are vectors etc). Is this just a coincidence resulting from the huge number of 3-variable equations in physics, or have the symbols intentionally been defined like this? Is there a deeper meaning? Are there other examples like this?

Answer 1

Just a coincidence. There are too many quantities and not enough letters. It probably does make a difference that the fields in which these two equations exist (material science and electromagnetism) are well enough separated that you typically won't see them both in the same papers or textbooks; if that weren't the case, people would start using different symbols. There are some examples where different fields use different notation for the same equation for precisely this reason.

Note that when I talk about two equations being the same, or different, I do it based on their meanings, not how they're written. So for example, your question is asking about two different equations that happen to share the same standard notation.

Answer 2

Yeah, that's just a coincidence. The easy way to see this is that $\epsilon$ is a relatively static property of a dielectric but a totally dynamic property of a stretching material.

Answer 3

Engineers created that problem. ;) (probably not) Many physics books use $Y$ for Young's modulus (Symon, Knight, Young & Freedman). Taylor's Classical Mechanics uses YM. Halliday, Resnick & -fill-in-the-blank- state that engineers use $E$.

I suspect that physicists started using $Y$ for exactly this reason: to highlight a difference in the meanings of $\frac{\sigma}{\epsilon}$.

From a different perspective, engineers write Hooke's Law as $\sigma = E\epsilon$, where $E$ is the proportionality factor called modulus of elasticity or Young's modulus. See a mechanics of materials text like Hibbler or Beer & Johnston.

Physicists rarely begin with $\sigma = E\epsilon_0$. That usually appears two or three lines down in a solution as an algebraic rearrangement.

Answer 4

Coincidence, nothing deep I'd say. Note that the equation representing the electric field modulus depends on the units you've picked and as such putting so much emphasis on the exact characters appearing in the eq. is senseless. Note that it's possible to form many physics equalities and equations involving 3 characters. E, epsilon and sigma are quite used. One could calculate the probability for this coincidence to have happened. I guess a full answer to your question could involve such calculations.