As we know, charged particles have polarity and like charges repel each other and unlike charges attract each other, and we have Coulomb's law to find that force. But how does it work? Does it work like gravity, like when two like particles are apart do they still repel? Is there any observational data relating to it?
[Physics] How do charged particles interact with each other
carrier-particleschargecoulombs-lawquantum-electrodynamicsvirtual-particles
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Now per QED, electrical charges interactions are effected by photons. Suppose you are one of the two charges. How do you know to attract or repel the other charge?
You want something that does not exist - intuitive picture of physical process within a theory which is a demonstration of how far can one go with mathematisation of experience and ignoring intuitive pictures.
To study quantum electrodynamics you have to concentrate on its computational algorithms and neglect intuitive pictures, to study intuitive pictures you have to neglect QED.
Both are a good thing to study, just do not expect it is easy to make them consistent.
So, on a fundamental level why does the law of charge work? What causes like to repel like and opposites to attract at the smallest level
You are really asking why like repels like and opposites attract at the smallest level.
Physics does not answer ultimate "why" questions, because it is a discipline which describes with mathematical models what is observed in nature. The models differ from maps because they not only fit existing data/measurements but are also predictive of new results of experiments and observations. Then the model can be used to answer why questions by how from one state another state can be predicted or described. The ultimate why is contained into the laws and postulates of the theoretical model, which are a distillation of observations/measurements or necessary to identify the mathematical functions with physical measurements .
In electromagnetism it was observations of how matter could be charged and of how charges interacted that developed into the law of Coulomb. This means that the existence of opposite charges assigned to particles is a given of nature, a law.
Coulomb's law, or Coulomb's inverse-square law, is a law of physics that describes force interacting between static electrically charged particles.
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The force of interaction between the charges is attractive if the charges have opposite signs (i.e., F is negative) and repulsive if like-signed (i.e., F is positive).
This was the classical macroscopic observation that is implicit in the laws and postulates of electromagnetism, i.e. the physics axioms that pick out from the infinity of mathematical solutions of the differential equations of the mathematical model those that describe nature and can predict new observations.
Once the microcosm started being explored classical mechanics and classical electrodynamics became inadequate to describe and predict behaviors. Quantum mechanics and special relativity were necessary to describe mathematically and predict results.
The laws of the classical regime are also laws of the quantum mechanical regime or can be seen to emerge from them. This is necessary because there should be a smooth continuity in the predictions of the solutions of the models in phase spaces where both views could be used to calculate and predict charged particle behaviors.
So the answer to your question of of "What causes like to repel like and opposites to attract at the smallest level" , i.e is because that is what measurements and observations say. The "how" is given by the corresponding mathematical theory of quantum electrodynamics
QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism giving a complete account of matter and light interaction.
Thus the "how" can be described mathematically, given the axiom that charges exist and like repels like and opposites attract.
Best Answer
We have frameworks in physics where questions can be answered: the classical electrodynamics framework, the quantum framework, the special and general relativity framework etc. Frameworks differ in the variable's range of validity, but blend smoothly in the overlap region.
In current day physics particle is a name given to electrons, muons etc, in the particle data table of the standard model of particle physics.. This is the quantum mechanics framework.
Your question is in the classical electromagnetism framework, so these are classically defined particles carrying charge .
yes, if by polarity you mean + or - charges.
Correct.
Yes, the field of each particle overlapping generates a repulsive force, following Coulomb's law, theoretically no matter how far apart they are. Note the term law. Laws are axioms in a physics theory, they tie measurements and observation to the mathematical model, in this case the 1/r^2 law.
A lot. Classical electrodynamics is a theory with Maxwell's equations that describes and predicts all possible situation macroscopically, and has not been falsified by measurements. This includes Coulomb's law.
Now let us go back to particles in the microscopic, quantum mechanical framework, like two electrons repelling each other :
The mathematical formulae represented by this feynman graph, for large distances will display Coulomb law behavior. At small distances it is complicated and needs the study of new mathematical tools as displayed by the graph above.