---
tag: DBQXRTC0VN
kind: concept
name: Logarithmic capacity
msc: [31A15, 30C85]
sources:
  - url: https://en.wikipedia.org/wiki/Conformal_radius
---

## Summary

The logarithmic capacity of a compact set $K$ in the plane measures how large $K$ is from the point of view of [[potential theory|9PM8Q2E30Q]].
It is defined from the least possible logarithmic energy of a unit charge distributed on $K$.
A disc of radius $r$ has capacity $r$, and a line segment of length $L$ has capacity $L/4$.
Sets of capacity zero, the polar sets, are negligible for many questions on harmonic and holomorphic functions.
Examples are removable singularities and the solvability of the [[Dirichlet problem|8D56GQ4AG8]].

## Explanations

### L1 human

Spread one unit of electric charge on a conducting plate shaped like $K$, in a two-dimensional world where the potential of a point charge is logarithmic.
The charge arranges itself to make the energy as small as possible.
The smaller that minimal energy, the larger the capacity.
A finite set of points cannot hold any charge with finite energy, so it has capacity $0$, while a disc of radius $r$ has capacity $r$.