# Origin of the coulomb and ampere

What is the historical origin of the size of the coulomb (and in turn, ampere)?

Currently (pardon the pun), the coulomb is defined in terms of the ampere. The ampere is in turn effectively defined by the vacuum permeability, which is exactly defined as $$\mu_0 = 4 \pi \times10^{-7} H / m$$.

According to Giancoli, this exact value of $$\mu_0$$ was chosen because it very closely matched the value of $$\mu_0$$ when measured with an independently defined ampere (which, I assume, was defined as $$1 C/s$$).

None of this, however, answers my question: why is the coulomb the size it is today, an outrageous size that makes it impractical for direct use?

Edit: I'm more interested in the historical reasoning that gave the ampere and coulomb the sizes they have today, not the current definitions.

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• Why do you say that the ampere is outrageously large? A common amperage for circuits in a house is 15 amps. – G. Smith Nov 4 '18 at 2:47
• "which an outrageous size that makes it impractical for direct use?" - Starting current for a typical car: 100 - 200 amps. A typical water heater uses over 20 amps when heating. A Tesla Model S battery system supplies up to roughly 1600 amps in Ludicrous mode. – Alfred Centauri Nov 4 '18 at 3:12
• Would History of Science and Mathematics be a better home for this question? – Qmechanic Nov 4 '18 at 9:42
• Err... the coulomb is outrageously large, not the ampere. Ninja'd the question to reflect this. – theunamedguy Nov 4 '18 at 15:10

my question: why is the coulomb (and ampere) the size it is today, which an outrageous size that makes it impractical for direct use?

The coulomb has that value because in the mid 19th century electrical engineers needed practical units for submarine cables and telegraphy.

In 1861 a committee of the British Association for the Advancement of Science was appointed to propose a system of units that included electrical and mechanical units. The committee defined two coherent systems for scientists, called electromagnetic and electrostatic units, and added several "practical" units that were decimally compatible with the electromagnetic units, for the electrical engineers involved in telegraphy and submarine cables.

In 1861 the electrical engineer Latimer Clark suggested to the committee that practical voltages were in the range of 1 to 10^6 volt, resistances of conductors and insulators were in the range of 1 to 10^8 ohm, and the smallest current was about 10^-3 ampere (translated to modern units). He proposed names like volt and ohm for these practical units, and the prefix mega for 10^6. The unit of charge was unimportant for submarine cables and telegraphy, so nobody cared about its practical value, it merely had to be coherent with the unit of current.

In 1881 the International Metre Convention adopted the practical units and their names: volt, ohm, and ampere. These practical units are the ones that scientists and engineers are using today.

Wikipedia’s “Ampere” article has a History section which explains how the ampere was originally defined:

“The ampere was originally defined as one tenth of the unit of electric current in the centimetre–gram–second system of units. That unit, now known as the abampere, was defined as the amount of current that generates a force of two dynes per centimetre of length between two wires one centimetre apart. The size of the unit was chosen so that the units derived from it in the MKSA system would be conveniently sized.”

Note that the numbers in this definition, other than 1, are 1/10 and 2, so the definition was reasonably natural. The coulomb was simply an ampere-second.

According to Ampere of wikipedia, SI defines ampere as follows:

The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to $$2\times10^{-7}$$ newtons per metre of length.

Generally, according to the Ampère's force law, for two straight parallel conductors of infinite length with current $$I_1$$ and $$I_2$$ and separation $$d$$, the force felt by a part of conductor with length $$\Delta L$$ is $$F = \frac{\mu_0 I_1 I_2 \Delta L}{2\pi d}$$

I would say the value of $$\mu_0$$ is defined by the definition of Ampere.