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Gauss Law Paper

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Composite Problems: Subtract voids By now, it may have become clear that there is a pattern to the solutions to Gauss’s Law for spheres, cylinders, and slabs. These can be summarized as follows: Assume a uniform volumetric charge density ρ in: a sphere and an infinite cylinder of radius w, and a slab with a thickness of 2w (so that w is like a radius, measured from the center), and assume field-points measured at r for the sphere, ℓ for the cylinder, and z from the central plane of the slab. The value of E for a field-point outside of a sphere, cylinder, or slab, respectively, is ρw³/3εr², ρw²/2εℓ, and ρw/ε. If the field-point is inside, they’re ρr/3ε, ρℓ/2ε, and ρz/ε. These can all be either memorized, or derived quickly if Gauss’s Law is understood. The appropriate values can be plugged in, and a solution quickly yielded. Suppose now that the goal is to find the E-field for a thick spherical shell— a sphere with a hollow center. Let its inner radius be wi, and the outer be wo, and the field-point at r is bigger than both. The easiest way to compose this conceptually is to have a solid sphere of radius wo and then subtract from it a smaller sphere of ...
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Tags: Gauss Law spheres Cylinders net electric flux Hyperphysics

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