📂Physics

Characteristic Length in Physics

Terminology ¹

In physics, the term characteristic length denotes the length that represents the scale of a physical system when describing a phenomenon.

Explanation

Characteristic lengths frequently appear when defining dimensionless quantities. Beyond the name, they are very useful for completing the expressions of most dimensionless numbers, so you should definitely be familiar with them.

However, as the vague wording suggests, the notion of “scale” can be hard to grasp, and commonly used examples are not consistent. For instance, if a system has volume $V$ and surface area $A$, it might seem natural to define the characteristic length as $L = V / A$. If the system is a cylinder with length $l$ and radius $r$, the characteristic length would be: $$ L = {\frac{ \pi r^{2} l }{ 2 \pi r l + 2 \pi r^{2} }} = {\frac{ r l }{ 2 l + 2 r }} = {\frac{ 1 }{ 2 }} \left( {\frac{ 1 }{ l }} + {\frac{ 1 }{ r }} \right)^{-1} $$

The reciprocal-average of $r$ and $l$! It may sound reasonable at first glance, but physics isn’t that simple. In fact, if you’re interested in a fluid flowing in a pipe, what matters is the pipe’s cross-sectional area, and the corresponding characteristic length is the pipe diameter $L = 2r$.

The point is that the characteristic length changes appropriately with the situation, so it is difficult to define it by a single formula.