Although I have never personally conducted an experiment, many online sources say that neutral and charged particles attract. However, under Coulomb's Law, I get a net force of $0$ since one of the $q$'s is $0$. Why is this?
[Physics] Is Coulomb’s Law not always accurate for neutral charges
chargecoulombs-lawelectrostaticsforces
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I am not quite sure which literature are you looking for, but it should be written in standard textbook. Anyway, you are right that it is not that useful except a formulation and exercise.
Usually, it is written in the following form. Given a charge distribution $\rho_{1}(\vec{\mathbf{r}_{1}})$ for object with volume $V_1$ and $\rho_{2}(\vec{\mathbf{r}_{2}})$ for object with volume $V_2$, we have the electric field: $$\vec{\mathbf{E}}(\vec{\mathbf{r}_{2}})=\frac{1}{4\pi\epsilon_{0}}\int_{V_{1}}\frac{\rho_{1}(\vec{\mathbf{r}_{1}})}{|\vec{\mathbf{r}_{2}}-\vec{\mathbf{r}_{1}}|^{3}}(\vec{\mathbf{r}_{2}}-\vec{\mathbf{r}_{1}})d\vec{\mathbf{r}_{1}} \tag{1}$$ and force $$\vec{\mathbf{F}}=\frac{1}{4\pi\epsilon_{0}}\int_{V_{2}}\int_{V_{1}}\frac{\rho_{1}(\vec{\mathbf{r}_{1}})\rho_{2}(\vec{\mathbf{r}_{2}})}{|\vec{\mathbf{r}_{2}}-\vec{\mathbf{r}_{1}}|^{3}}(\vec{\mathbf{r}_{2}}-\vec{\mathbf{r}_{1}})d\vec{\mathbf{r}_{1}}d\vec{\mathbf{r}_{2}} \tag{2}$$
Because of practicality, it is not used that often:
- It is hard to apply on non-fixed charge distribution. For example, for conductor, the charge itself would change as two object approaching. This internal dynamics is not captured in Eq (2).
- When the objects are far away, it is just like a point charge. Why do you want to do the cumbersome integration?
- There are better tools to handle this situation: Multipole expansion. As two objects become closer and closer, you include the monopole first (i.e. a point charge), then dipolar, then quadrupole... This expansion is very systematical and have good physical meaning.
- People care more about electric field rather than force, as it is more fundamental, so the Eq (1) is more emphasized than Eq (2).
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
Coulomb's Law describes the electric force between pointlike charged objects.
Things are more complicated for composite particles, which may be overall neutral, but which are composed of charged constituents. An atom has the positively charged nucleus, surrounded by a negative electron cloud. If the nucleus is slightly off center, then the forces it exerts do not need to cancel out the forces exerted by the electrons. The resulting forces tend to be much weaker, but they do exist.
When quantum mechanics is taken into account, neutral composite particles tend to attract one another; this is known as the van der Waals force. The reason is there are quantum fluctuations in the relative positions of the charged constituents of one particle; these fluctuations then exerts forces which slightly rearrange the constituents of another overall neutral atom. The rearrangement is in such a way as to make the attractive forces between the oppositely charged parts of the two slightly stronger than the repulsive forces between the like charged constituents.