I've just learned about the statcoulomb, which is basically a way to express charge when we didn't have units of charge (I think), with the definition $$1 \mathrm{statC} = 1 \mathrm{dyne}^{1/2} \mathrm{cm} = \mathrm{cm}^{3/2} \mathrm{g}^{1/2} \mathrm{s}^{-1},$$ but I don't understand where this comes from. I've read in various texts, and in all of them they explain the correspondencies and why this definition works, and I understand it, but I don't understand it conceptually, I mean, how can something like a Coulomb, which for me it couldn't be more far from units of length, mass and time, be explained in terms of these (not entirely, I know, since a Coulomb is different from a statcoulomb, but with a very close relation).
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$\begingroup$ You wrote "I've just learned about the statcoulomb, which is basically a way to express charge when we didn't have units of charge (I think), with the definition...". I think you are historically wrong. There is no dfiference between the definition of statC you report and the definition of newton as "the force which causes a mass of 1 kg to accelerate at 1 m/s^2." See also my folloeing comments. $\endgroup$– Elio FabriCommented Oct 28, 2018 at 14:34
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As you probably know, it comes from Coulomb's Law written without any constants: $F=q_1q_2/r^2$. Since we've already defined force and distance, the units of charge are fixed. There's nothing more to it.
Well, almost. There are at least four "cgs" systems of units: those based on magnetic force, those based on electric force, with and without a factor of $4\pi$. While these systems are good for theoretical developments, they are very confusing for experimental work, because devices generally display SI units. And confusion exists when comparing equations in the various cgs systems.
And then there is the conceptual problems, such as you are having. There's nothing to conceptualize. It's just a unit of charge.
It's little wonder that cgs systems are relegated to specialty fields.
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$\begingroup$ Yep. My general advice for understanding the statcoulomb is "don't". $\endgroup$ Commented Oct 26, 2018 at 23:48
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$\begingroup$ @garyp You forgot Gauss system (OK, you wrote "at least"). It's true that devices display SI units. I think instrument makers are obliged to do so, since SI is internationally recognized, for legal purposes too. This has nothing to do with physics, however. $\endgroup$ Commented Oct 28, 2018 at 14:35
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$\begingroup$ @Emilio Pisanty I don't agree. I don't want to enter the unfinished discussion about units and so on, but I want to stress one point. There's nothing of physically meaningful in the choice of one system of fundamental quantities or another, and in the ensuing choice of units. $\endgroup$ Commented Oct 28, 2018 at 14:38
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$\begingroup$ Neither do I agree with OP's idea: "... a Coulomb, which for me it couldn't be more far from units of length, mass and time". Neither less nor more than a newton. (BTW, in SI the unit of charge is to be written without capital initial. A capital letter is required when abbreviation is used: C, like N, like A or F, or J.) All this is just matter of convention, and choices may depend on several reasons: practical, historical... $\endgroup$ Commented Oct 28, 2018 at 14:41
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$\begingroup$ @ElioFabri SI units does have a physically suggestive meaning. $D=\frac{q}{4\pi r^2}$ is the ratio of the charge to the area of the sphere surrounding it. $\endgroup$– garypCommented Oct 28, 2018 at 19:01