Particle Charge Simple question that I've never heard a satisfactory answer for from a physics teacher: what exactly is a 'charge' of a particle?
Most teachers I've heard from define charge in one of two ways:

*

*In terms of its opposite (so you define a negative charge of an electron as the opposite of a proton's positive charge)

*In terms of its effects (i.e. a charge can attract or repel other charges)
My problem with those definitions is that it seems to me in either case you're defining a charge in terms of itself. For instance, the first definition is like saying cold is the absence of heat, and the second definition is like saying heat flows from hot objects to cold objects...but neither actually explains what heat is.

Is there a way to define what the charge is without referencing either?
 A: Here is another question: What is an electron?
An electron is a particle with certain properties. It has mass $m_e$, charge $q_e$, and spin $1/2$.
So what is mass? If a force is applied to an electron, the mass tells you how quickly it accelerates. $a = f/m_e$. If the electron is near the Earth, it tells you how strong the force of gravity is. $f = G \space m_{Earth}m_e/r^2$ If you convert the electron to energy, it tells you how much energy you get. $E = m_ec^2$. That is, mass is part of the math we use to describe the behavior of the electron, to predict outcomes of experiments we can do with electrons.
After a while, we get used to the idea of mass. We develop intuition about it. It is how "heavy" the electron is. It is the "amount of stuff" in the electron. It is the "source of gravity" of the electron, tiny though that is. These intuitions aren't strictly physics, but they help us understand.
Charge is similar. If you put two electrons near each other, they repel. An electron and proton attract. The force is $f = 4 \pi \epsilon_0 q q_e/r2$. If you put a moving electron in a magnetic field, the electron is deflected from a straight path. $\vec f = q_e \vec v \times\vec B$.
Intuitively, charge is the "source of electric and magnetic force" of the electron. It is the "strength of response" of the electron to other electric and magnetic forces.
Physics is a mathematical description of the universe. Physics isn't real in the same sense that the universe is real. Physics is math. What is fundamentally real is the particles and such in the universe, together with repeatable patterns of behavior exhibited by those particles.
The purpose of physics is to describe the behavior of the universe in the form of mathematical laws. And given those laws, predict the behavior of the universe.
Part of what we need for the laws is properties of particles like mass, charge, and spin.
You might be looking for an answer about charge that describes the mechanism how it makes the electron generate and respond to electromagnetic forces. Physics doesn't have answers for that. An electron has no internal workings we know of. It just is. It just has innate behaviors.
Protons are more complex. They are made of quarks. But that just pushes the question down a level. Quarks have no internal working we know of. They just are.
You can often describe complex laws in terms of simpler laws. But when you get to the most fundamental laws, you can go no further. We just look at the universe and describe it as best we can. Charge is a number that helps us do that.
A: Physics uses mathematics. Mathematics is a discipline where everything can be defined and accurately predicted , and questions within a purely mathematical theory  are answered by reference to theorems, and theorems by reference to the axioms of the mathematical theory. A why question in  a purely mathematical theory ends up on the axioms, and the answer then becomes "because of the axioms", otherwise it would be a different theory.For example for flat geometry one uses as axiom "two parallel lines do not meet". If they met, it would be a different geometry, for example spherical, as used to map the earth..
Now in order to use a mathematical system for physics, and pick up the correct solutions from the multitude of possible mathematical ones,  map physics data and , important, predict new data and observations, one needs extra axiomatic statements, derived from observations. These are called "laws", "principles", "postulates". In addition various assigned quantities to observations are also axiomatic. Why questions end up on these defining axiomatic statements.
The standard model of particle physics has the quantum mechanics postulates, and an axiomatic definition of particles in order to successfully model the data. There are a lot more  numbers defining particles than charge. All are derived from data.
Charge was seen from ancient times, and in classical electromagnetic theory a particle can have a charge, but it was only with the study of particle physics that it was found that  the electron has a definite charge in the macroscopic units of charge seen here ( we tend to call the charge of the electron -1, but it has to be multiplied by the units in the table). So it is an observational fact that elementary particles have charge.
