The roundabout definition of electric charge

Question

In the book called Electricity and magnetism by Purcell, in page-240, he writes that Q in a surface is defined as
$$ Q = epsilon_{o} int_{partial S(t)} vec{E} cdot vec{dA}$$
Now, I'm quite confused with this definition because the Electric field of charge distribution is defined using charge distribution and now he goes back and defines charge distribution using electric field... Isn't this circular reasoning?? I don't particularly think this is a convincing definition of charge
Just above the line of equation 5.2, he writes: " We define the amount of charge
inside S as $epsilon_{o}$ times this integral:"

Claudio Saspinski · Accepted Answer

I think it can be justified because we can only know that there are charges there due to the effect in other charges. And that effect is expressed mathematically by the electric field.
So, using test charges outside at several distances, it is possible to evaluate the field. While the charges are the source of the field, and on this way more fundamental so to speak, they are only known due to its effects, due to the field they create.
An example where we could imagine it could also be done, let's take mass as the source of the gravitational field. According to Newton's gravitational law, there is an acceleration outside any mass $M$ given by: $$|mathbf a| = frac{GM}{r^2}$$
The reason for not defining mass as $$M = frac{ |mathbf a|r^2}{G}$$ is that mass, for all everyday objects, has other and much bigger effect than the gravitational one. It is the inertial property, expressed by the relation: $mathbf F = mmathbf a$.
So, the gravitational effect of mass is only meaningful for huge masses, in the scale of planets. And its main effect in the normal life has nothing to do with its gravitational field.
It is not the case of charges and electric field. The latter is the sole effect of charge, and can be used as its definition.

Han-Kwang Nienhuys · Answer

This is a guess about what the author was thinking.
We have the fundamental SI units for length (m), mass (kg), and time (s), which were originally defined as one 40-millionth of the circumference of the earth, the mass of 0.001 m3 of water, and 1/86400 of a day. This is the MKS system. There is no natural way of integrating electromagnetism in this MKS system of units and there are actually several conventions, the most common of them being MKSA (A for ampere), and the other CGS-Gaussian (cm, g, s, and a bunch of obscure units for electromagnetism). The difference is not just in the names of the units, but also in the equations describing electromagnetism. For example, in CGS-Gaussian, the equation would read
$$ Q=4pi int_{partial S} vec{E}cdot dvec{A}, $$
which has a factor $4pi$ instead of $epsilon_0$; there is no such thing as $epsilon_0$ in CGS-Gaussian units; the unit of charge is equal to $mathrm{g^{1/2},cm^{3/2},s^{−1}}$ and the unit of $E$-field is $mathrm{g^{1/2},cm^{-1/2},s^{−1}}$. So, in a sense, you define the concept of charge by the equations.
But I wouldn't go as far as defining charge from the surface integral of the $E$ field, because it would require that you have a way to quantify electric fields without involving electric charges. You noted this yourself.
A more meaningful definition of charge is to start from the ampere (1 coulomb is 1 ampere second), along with the original SI definition that links the ampere to the MKS system:

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 2×10−7 newtons per metre of length.

(The present SI definition defines the coulomb as a fixed multiple of the elementary charge.)
A more down-to-earth explanation of choice of words is that the authors were sloppy. Maybe the book was originally (in 1965) written for CGS-Gaussian units and the phrase made more sense in the original explanation.

knzhou · Answer

It’s not circular at all. The early chapters are concerned with charges at rest; here their electric fields are given by Coulomb’s law.
The chapter where this new definition appears is about the fields of moving charges, a topic which has not yet been covered. In this more general context, Coulomb’s law doesn’t work anymore, so we start over with new definitions. Before, it was trivial to define charge; now, it isn’t, so it’s useful to start with a definition of it.

The roundabout definition of electric charge

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