Electric field (article) | Electrostatics | Khan Academy

There are two ways to think about charge. We know that charge is the property of two atomic particles, electrons and protons. This makes it convenient to think about charge as particles, or like a bunch of sand. You can count sand particles (if there are not too many). Coulomb’s Law treats charge this way, there’s a q1 and a q2.

Another way is to think of charge as a continuous substance, like peanut butter. Peanut butter isn’t a collection of particles, it’s something different. You charge something by slathering it with peanut butter charge. The charge is uniformly distributed throughout the peanut butter.
If you see a problem statement like “assume a uniformly charged rod,” that’s an example of the continuous peanut butter version of charge. Continuous charge will include a density specification like 2 coulombs per meter, or 3 coulombs per cubic inch.

If you are presented with a problem based on peanut butter charge you have to figure how to apply particle-based Coulomb’s Law. In this blob of charge we have to somehow identify a charge particle. The trick is to use calculus to focus down on a tiny tiny bit of the charged structure, a bit so small it can be considered a particle.

So in the article you see the equation for the electric field from multiple charges

F = 1/4pieo SUM (q_i/r^2)

In peanut butter charge q_i becomes the differential charge dq, and the SUM turns into (evolves into) an INT (integral).

F = 1/4pieo INT (dq/r^2)

These two equations mean the same thing. In the second we rely on calculus notation to do the bookkeeping for adding up all those infinitesimal dq’s.

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