Coulomb and Gauss’ Laws
~ with static
• We revisited the two Maxwell’s equations involving curl and divergence of E
fields in vacuum. Our knowledge of curl and divergence properties gives these equations a
physical feel. One equation says that electric fields don’t curl (they form half loops starting
and ending on point charges). The other says that the divergence is proportional to the
volume charge density, which using divergence theorem is a differential representation of
Gauss’ law (flux through a closed surface is proportional to total charge enclosed).
• Gauss’ law is useful for simple geometries with symmetry, where one can choose Gaussian
surface such that (a) their geometries are simple and surface areas are easy to evaluate,
and (b)where the field lines are directed parallel or antiparallel to the normals of every
surface, allowing us to simplify the dot products within the integrals. Simple applications
give us the field due to a point charge ( 1/r2 ), line charge ( 1/r), etc. Each power can be
justified by multiplying Coulomb contributions from each and the number of charges.
• For general geometries, one needs to use Coulomb’s law, along with superposition principle. One can use the expression proportional to ~r/r 3 to add contributions. This can be
generalized to continuous distributions over line, surface or volume, such as a ring or disk
of charge.
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