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This might be a strange physics question, but I am curious about it. Are there any physics equations that involve exponents $\geq$ 5?

There are plenty of equations with square exponents. Inverse square laws are the most obvious to me, like the universal gravitational force law. $$ F = \frac{GMm}{r^2}$$

I've seen some equations with an exponent of 3, like power to overcome drag in a fluid:

$$ P = \frac{1}{2}\rho v^3 A C_d $$

And also various cube-square laws like the semi-major axis related to the orbital period:

$$ a^3 = \frac{GMT^2}{4\pi^2} $$

I've also seen an equation with an exponent of 4. It's the Luminosity equation for black bodies, usually used for star luminosity: $$ L = \sigma A T^4 $$

There are also various beam-deflection formulas involving the 4th power of length, like a simple beam supported at both ends:

$$ \delta _{max} = \frac{5qL^4}{384EI} $$

But that might be considered more engineering than physics.

Anyway, I have never seen a physics equation involving an exponent of 5 or more. Do they exist? Is there some reason why they are super-rare or non-existent?

Admittedly, there is some arbitrariness in this question. For example, we could take any law we want, like Hookes Law, and take both sides to the 5th power...

$$ F = -kx $$

$$ F^5 = -k^5x^5 $$

But that's way too arbitrary. I guess the best way to avoid that is to say we want a physics equation in its most simple or most useful form.

Also, you could say that the hypervolume of a cube in 5 dimensions is equal to $x^5$. That is a little too arbitrary too. Not even sure if that's physics or just plain geometry.

Qmechanic
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DrZ214
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  • A spring force can be proportional to $F=k,x^5$ – Eli Jan 11 '20 at 17:07
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    https://en.wikipedia.org/wiki/Lennard-Jones_potential there certainly are, but this has been asked before https://physics.stackexchange.com/q/508797/23615 – Triatticus Jan 11 '20 at 17:07
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    One of the critical exponents in the 2D Ising model is 15. – knzhou Jan 11 '20 at 17:08
  • Does Cauchy's relation count? – Sam Jan 11 '20 at 17:23
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    Electric fields of octupole and higher multipole charge distributions have inverse distance exponents $\ge 5$. Single-particle nuclear model multipole gamma energy transition probabilitiy in nuclei of quadrupole and higher are proportional to $E^5$ or greater. – Bill N Jan 11 '20 at 17:52
  • The Planck law wrt lambda^5 – Poutnik Jan 11 '20 at 17:59

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