Are there any physics formulas with high/large exponents?

Question

Most physical laws seem to only have low integer exponents for their variables - in my experience I've never seen a physical law containing variables raised to a power greater than 3 or occasionally 4. Are there any physical laws containing variables raised to large integer exponents (i.e. powers of 5 or higher)?

Answer 1

The tangential force in the skin of an inflated balloon is calculated by an equation that includes the sixth power of the ratio of its original to expanded radius. See the Wikipedia article on the two balloon experiment https://en.wikipedia.org/wiki/Two-balloon_experiment

Answer 2

One of the most common reasons one finds that physical laws don't contain higher-order exponents is because of either linearity or locality approximations, which manifest the same mathematical consequence. There are other possible reasons for low-order exponents, such as the 3+1 dimensionality of our universe, but in my opinion the first two I mention are the biggest.

To demonstrate, consider some relationship between two physical variables of interest, $\lambda$ and $\mu$ of the form $\lambda = f(\mu)$. If the relationship is linear, then there's no need to overthink it: $\lambda = c \mu$ for some constant, and no higher-order powers are needed.

If the relationship is nonlinear however, we can either directly model the nonlinear relationship or spot some sort of workable approximation for the relationship around some neighborhood of $\lambda$'s or $\mu$'s for which we can obtain experimental validation. In this second scenario, one constructs a local approximation out of a Taylor expansion for some value $\mu_0$:

$$\lambda = f(\mu_0) + \frac{df}{d\mu}|_{\mu_0}(\mu-\mu_0) + \frac{1}{2}\frac{d^2f}{d\mu^2}|_{\mu_0}(\mu-\mu_0)^2 +\ ...$$

Because the higher-order terms are multiplied by factors of $(\mu - \mu_0)^n$, the lower order terms always dominate in the region of interest, which means they are rarely considered.

In the situation where one now models the full nonlinear relationship directly, it's very rare to obtain a function that is a high-order monomial—simply because it's rare to get any kind of finite-order polynomial directly in such a situation. (Note the ubiquity of exponentials and sinusoids in fully nonlinear physical relationships!)

That being said, your best bet of finding high-order polynomials will result from such fully nonlinear relationships: the Stefan-Boltzmann law, for example, generates a host of nonlinear terms as a direct result of using fully nonlinear forms of the density of states & the partition function for a bosonic gas:

$$j = \frac{2\pi^5 k^4}{15h^3c^2} T^4$$

Only the $\pi$ cracks your $n\geq5$ limit here, but hopefully you see my point. Try and see what happens if you derived the equivalent law in, say, a $6+1$ dimensional universe!

QUASI-SPOILER: In 5-D, you'd get that $j = \frac{8\pi^8 k^6}{63h^5c^4} T^6$. Plenty of big exponents here!

Answer 3

The exponents of laws become arbitrarily large when you start getting in to Multipole expansions. For a $2^N$-pole potential, the potential will fall off, to leading order, like $r^{-1-N}$ as $r\rightarrow\infty$ (with the force falling off like $r^{-2-N}$). This falloff goes even faster when you consider induced multipoles. For example, when an otherwise symmetric charge, like a ground state hydrogen atom, responds to an electric field by producing a multipole field of its own (part of the Lennard-Jones potential, for example, is the atoms mutually inducing dipole moments in each-other).