Let's intuitively learn two Maxwell's equations - Gauss's Law - intuitively. And solve in minutes, what Newton couldn't in years.
The first two Maxwell equations - The Gauss law - comes directly from Coulomb's law. That's right, by rearranging terms and doing some clever arguments, we can logically show that the flux through a closed surface must indeed equal the total charge enclosed by the surface. And in doing so, we will uncover why we define electric flux as the dot product of electric field and the area in the first place.
Then, we apply this to find the electric field due to a sphere of charge, outside it. Newton did a similar calculation - finding the gravitational field due to sphere of mass - using integrals. You can find the proof of that online Search for Shell theorem.
But, using Gauss's law, we can find the answer much quicker.
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