The maths behind the trans-Atlantic telegraph cable
In the mid-19th century, the most sought-after feat of engineering and industry was a trans-Atlantic telegraph cable. But when the first successful cable was laid between Britain and America in 1858, the speed of transmission was only about 0.1 words per minute. This post will be an analysis of Thomson's 1855 paper "On the Theory of the Electric Telegraph", published in the journal of the Royal Society of London. Derivation of the PDE Thomson starts by defining the variables that he uses throughout the work. I shall define the same variables using more modern lettering conventions. Let $C$ be the capacitance of the wire, $R$ the resistance of the wire, $V$ the potential at a point $P$ on the wire at a time $t$, and $I$ the current at the same point in the same instant. The charge $Q$ at $P$, called "quantity of electricity" by Thomson, is given by \[Q = VC = It\] Thus, in an infinitesimal length $dx$ of wire at $P$, we have a charge of $VCdx$ and, i
