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Well that depends. As always, you should be very careful with such reasoning as Due to the fact that gravity is related to the square of the distance should not the gravitational sum of every particle exceed the force when calculated by the center of mass. because this is a problematic statement. In general your (Newtonian) gravitational potential is ...

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The principle in general is called superposition in physics or linearity in mathematics. It is very useful when you want to study a system for that system to be approximately linear. What linearity is, in a more general context. Here "linear" is the property of a function (or an operator, or whatever) to distribute over addition. So for example if you have ...

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"I find lots of solutions on the internet that say you can replace the cavity with a negative density, why?" Because they use a trick to calculate the potential easier. They assume that the empty hole is neutral, but composed of a positive charge density equal to that of the sphere plus a negative charge density of the same amount. In this way you can ...

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I think we should start with the local form of Gauss's law $\nabla.\vec{E}=\rho$ Now $$\int \nabla.\vec{E}\,dv=\int \rho\,dv$$ Using Gauss's divergence theorem we have $$\int\vec{E}.\vec{ds}=q$$ I assume $\epsilon_{0}$ to be 1 but you can always put that back into this. I think this way of looking at it does not assume any coordinate dependence.. Ofcourse ...

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When the point charge is not at the center of the sphere, the electric field lines will not intersect the sphere at right angles. Consequently, there is an initial component of electric field along the surface of a conductor. We know this results in a force on the charge carriers inside the conductor, and these charge carriers will re-arrange until the ...

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If $c$ is unknown, then you don't know it and you'll have to leave it (either directly or in another form, like $\rho$ or $Q$) as a variable in your expressions. So indeed you can either replace $c$ with $\rho/r^2$ or $Q / (4\pi R^5/3)$, but you cannot eliminate it completely unless they ask for some problem with a very fortuitous cancellation.

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