I’m writing my final paper on the 1858 transatlantic telegraph. One of the important moments in the story is the controversy over Thomson’s Law of Squares, which describes the behavior of signals in submarine telegraphs. Thomson discovered the law in 1855, and said that a longer cable would also have to be wider. But the engineer designing the cable, Wildman Whitehouse, said it should be smaller. In the end, Thomson was right, Whitehouse was wrong, the cable failed, and Whitehouse was fired. I didn’t have space in my paper to discuss the theory in detail, but it’s a fascinating story in its own right. If I were to start over, I might write a paper just on the law of the squares.
Submarine cables, unlike landlines, are surrounded by water, which makes them act as giant capacitors. They form a low pass filter, smearing and distorting the telegraph signal. The Victorian electricians called it “retardation of signals”. Faraday described the effect in his 1854 paper “On Electric Induction – Associated cases of Current and Static Effects.” He correctly observed that the slow charging and discharging in submarine cables was caused by charge building up on the outer surface of the wire and the inner surface of the water, forming a capacitor. Then the history gets tricky. According to several of my secondary sources, Faraday said that a thicker cable will have a greater retardation effect. I couldn’t find that in any of his writings, but it does seem to fit in with what he thought; and the original source for this historical claim is Charles Bright, who was in no hurry to defend Whitehouse. It would be a great topic for further research, but as far as I can tell, Faraday agreed with Whitehouse.
Faraday’s (alleged) prediction makes sense at first – a bigger wire would indeed hold more charge. But Faraday wasn’t so good at math. An intuitive, simplified explanation: If you scale up a cylindrical capacitor, keeping all ratios the same, the surface area of the wire increases in the same proportion as the width of the dielectric barrier, so the capacitance (which is proportional to area divided by separation) doesn’t change. Thomson solved the problem more rigorously by calculating the capacitance (“electro-statical capacity”) per unit length of a cylindrical cable. His formula was correct, though we use different symbols and scalars today:
I is the “specific inductive capacity”, a constant based on the material, like the modern dielectric constant. This formula says that the capacitance depends on the ratio of the inner and outer radius. So changing the width of the cable – and keeping the insulation in proportion – has negligible effect on the capacitance. What does change is the resistance, which varies inversely as the square of the radius. Almost all engineers knew that.
Thomson’s big breakthrough was finding a formula for the retardation effect. As a modern engineer might intuitively expect, it involves an RC time constant; that is, the rise time is proportional to the product of the wire’s resistance and capacitance. But a telegraph has distributed resistance and capacitance, and they won’t know about Thévenin equivalence for another few years, so Thomson couldn’t just solve a first order circuit and be done with it. Instead, the thermodynamics expert took a heat diffusion equation and adapted it for circuits. He used the charge (“quantity”) passing through a differential length as a function of time to produce the differential equation:
where k is the resistance per unit length of the wire, c is the capacitance per unit length, and v is of course the potential. This is a classic diffusion equation, one I spent weeks studying in my entry-level differential equations course last semester. Thomson didn’t have the tools we have today (the other big player in transmission theory, Oliver Heaviside, would develop those tools later in the century), so he used Fourier analysis to solve it. Fourier mathematics was relatively new at the time, but it’s widely used today, especially by electrical engineers doing signal and system analysis. I’ve taken two courses devoted almost entirely to it! He found that, under certain assumptions, the cable signal decays with a time constant:
This is all to say that the retardation effect, as measured by the time constant of the signals, varies:
- Directly as k, the resistance per unit length of the copper wire. But k itself varies:
- Inversely as R2, where R is the radius of the wire.
- Directly as c, but c is a function of materials and of geometry. They could improve it by making thicker gutta-percha.
- Directly as l2, which is the law of squares itself.
Therefore, of the parameters that telegraph engineers can conveniently change:
To maintain the same time constant, therefore, the telegraph cable’s width would need to be increased in proportion to its length. That is Thomson’s law of squares. Thomson’s math was as correct as it was beautiful. He didn’t have experimental data to corroborate his claims, though, so no one was convinced until many years later.
