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Why that exact number of electrons in one coulomb? who decided it? there is nothing wrong with the number, it just seems slightly messy. Why didn't the scientific community just settle on an easier number, such as $1\times10^{-19}$ for example?

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The charge of $1C$ was derived from the definition of Ampere. If you look at the SI units, you'll check that, surprisingly, intensity of current is a basic unit, whereas charge is a derived quantity. This is a bit weird, because charge is seen as "more fundamental" than current, current is "charge per unit time".

So why is it? Because measuring the charge of one electron is very hard (electrons are extremely tiny), whereas currents are easily measurable.

Consider two straight and infinite parallel wires. The force exerted between the two per unit length is

$$f=\frac{\mu_0 I_1 I_2}{2\pi r}$$

Where $I$ are the intensities, $f$ is the force per unit length, $r$ is the distance between the wires and $\mu_0$ is a constant of known value. If we make $I_1=I_2=I$, we get

$$ f=\mu_0 I^2 / 2\pi r$$

So $I=\sqrt {2\pi r f /\mu_0}$

If we introduce the SI units: $r=1m, f=1N/m$, we get the definition of one ampere.

And then we define 1 coulomb to be $1C=1A\cdot 1s$.

So the value of $1C$ was derived first. Then, Millikan discovered how many coulombs was the charge of an electron.


EDIT for clarifiaction:

This is the historical process that led to the definition of one coulomb of charge.

The Ampere definition has recently been modified.

This answer explains the process for which: 1) The formula of the magnetic force between two straight current-carrying conductors was found. $f\propto I^2$ 2) This was used to define the unit of intensity of current. 3) Then the defintion of charge is striaghtforward. $1C=1A\cdot1s$. It was done like this because measuring currents is easier than measuring charges.

4) Millikan found the charge of the electron. He did it using the existing unit: coulombs. It happened to be $\sim 1.6\cdot10^{-19}$.

5) The definition of Ampere has recently been changed, in order to make it less dependent. However, this change has been such that the figure does not change, because we do not want all books and instruments to become wrong.

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    $\begingroup$ This is inaccurate. This answer describes the state of affairs through to early 2019, but as of last month, the ampere is no longer defined this way. See What are the proposed realizations in the New SI for the kilogram, ampere, kelvin and mole?, What is a base unit in the new SI, and why is the ampere one of them?, and their multiple Linked questions, for more details. $\endgroup$ – Emilio Pisanty Jun 5 at 14:54
  • $\begingroup$ Nice links, but I didn't say this is currently like this (if I said it, sorry because I didn't mean to). What I intended to say is how it was discovered. The OP asks how decided to set that number. Well, I'm exposing the historical reason why the number ended up like it is. The recent change maintains these figures because it'd be a mess to change it all now, and that's because it worked like this. However, it's true that this question could have been posted in hsm SE. $\endgroup$ – FGSUZ Jun 5 at 17:45
  • $\begingroup$ Absent such a disclaimer, this answer is still inaccurate. If you don't explicitly say that the system you're describing is obsolete and is no longer operational, then you're not answering the question, which is clearly about the redefinition. As it currently stands, this answer is wrong. $\endgroup$ – Emilio Pisanty Jun 5 at 17:58
  • $\begingroup$ @EmilioPisanty I'm sorry sir, but I fail to see why it is wrong. The OP asks why it is that number. And my reasoning is: 1) The definition of 1 ampere based on the magnetic force. 2) Definition of $1C4 using that Ampere. 3 ) Millikan found the electrno charge in terms of the unit what was in use. 4) New redefinitions should preserve that number (regardless on their new basis) because otherwise all instruments and measurements should be redesigned. Is this wrong? On the other hand, if you think that I should say this more explicitly, then it's fine and I'll edit my answer. $\endgroup$ – FGSUZ Jun 5 at 19:36
  • $\begingroup$ This answer treats the force-based definition of the ampere as if it still were in use, and it is not. That makes it inaccurate and misleading in its current form. The edits you propose can definitely solve the problem, but they won't if you refuse to implement them. I don't see what's hard about this. $\endgroup$ – Emilio Pisanty Jun 6 at 10:15
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I edited the question to be of the right form. The ampere is now defined by fixing the numerical value of the elementary charge (the charge of an electron or proton) in the International System of Units to be exactly equal to $1.602176634\times10^{-19}$ coulombs.

So why not a nice round number, like $10^{-19}$ coulombs, or a nice round number such as 6000000000000000000 ($6\times10^{18}$) for the number of electrons in a coulomb?

The answer is simple: Doing so would break everything electronic. Old ammeters and new ammeters would give different readings. Replacing an old 20 ohm resistor with a new 20 ohm resistor might fry a circuit. Whenever a metrological standard is updated, the new and improved version has to be consistent with the version it is replacing (and it has to be improved as well). The value of $1.602176634\times10^{-19}$ ampere-seconds (i.e., coulombs) for the elementary charge is consistent with the old definition of the ampere, to within experimental error.

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The charge on the electron, $1.602176634×10^{−19}$ coulomb which is $6.24509 . . .\times 10^{18}$ electrons per coulomb, was chosen to make the new definition of the ampere be in terms of coulombs and seconds but not kilograms and metres, as close as possible to the less precise and less reproducible old ampere definition which required the measurement of a force ie in terms of kilograms, metres and seconds.

This also meant that except for very precise measurements instruments calibrated before the new ampere was defined would not have to be recalibrated.

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