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  "body": "\n## Definition\n\nAn *electrical network* is a pair $(G, c)$ of a connected [[graph|00C0BR7D7H]] $G = (V, E)$, in which every vertex has finitely many neighbours, and a function $c \\colon E \\to (0, \\infty)$.\nThe number $c(e)$ is the *conductance* of the edge $e$, and $r(e) = 1/c(e)$ is its *resistance*.\nFor $x, y \\in V$, write $c(x, y) = c(\\set{x, y})$ if $\\set{x, y} \\in E$ and $c(x, y) = 0$ otherwise.\nA *voltage* is a function $v \\colon V \\to \\R$.\nThe *current* of $v$ is the function on ordered pairs of neighbours given by Ohm's law: $i(x, y) = c(x, y)\\big(v(x) - v(y)\\big)$.\n\n## Remarks\n\nThe current is antisymmetric: $i(y, x) = -i(x, y)$.\nThe net current out of $x$ is $\\sum_y i(x, y) = \\sum_y c(x, y)\\big(v(x) - v(y)\\big)$.\nKirchhoff's node law at $x$ says that this net current is $0$, which holds exactly when $v$ is [[harmonic|WRY0M7SW59]] at $x$.\nExample: the path $a - m - b$ with two edges of conductance $1$ is a network.\nThe voltage with $v(a) = 1$, $v(m) = \\tfrac12$, $v(b) = 0$ is harmonic at $m$, and it sends the current $\\tfrac12$ along each edge from $a$ to $b$.\nNon-example: giving an edge conductance $0$ is not allowed; such an edge carries no current and should be removed from the graph instead.\nKirchhoff's cycle law, that voltage drops add to $0$ around each cycle, is automatic here because the current comes from a voltage.\n",
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