Simple Nature - Light and Matter

with charge, change the Coulomb constant k to the gravitationalconstant G, and insert a minus sign because the gravitational fieldsaround a (positive) mass point inward.Gauss’ theorem can only be proved if we assume a 1/r 2 field,and the converse is also true: any field that satisfies Gauss’ theoremmust be a 1/r 2 field. Thus although we previously thought ofCoulomb’s law as the fundamental law of nature describing electricforces, it is equally valid to think of Gauss’ theorem as the basic lawof nature for electricity. From this point of view, Gauss’ theorem isnot a mathematical fact but an experimentally testable statementabout nature, so we’ll refer to it as Gauss’ law, just as we speak ofCoulomb’s law or Newton’s law of gravity.If Gauss’ law is equivalent to Coulomb’s law, why not just useCoulomb’s law? First, there are some cases where calculating afield is easy with Gauss’ law, and hard with Coulomb’s law. Moreimportantly, Gauss’ law and Coulomb’s law are only mathematicallyequivalent under the assumption that all our charges are standingstill, and all our fields are constant over time, i.e., in the studyof electrostatics, as opposed to electrodynamics. As we broadenour scope to study generators, inductors, transformers, and radioantennas, we will encounter cases where Gauss’ law is valid, butCoulomb’s law is not.10.6.6 ApplicationsOften we encounter situations where we have a static chargedistribution, and we wish to determine the field. Although superpositionis a generic strategy for solving this type of problem, if thecharge distribution is symmetric in some way, then Gauss’ law isoften a far easier way to carry out the computation.i / Applying Gauss’ law to aninfinite line of charge.Field of a long line of chargeConsider the field of an infinitely long line of charge, holding auniform charge per unit length λ. Computing this field by bruteforcesuperposition was fairly laborious (examples 10 on page 574and 13 on page 580). With Gauss’ law it becomes a very simplecalculation.The problem has two types of symmetry. The line of charge,and therefore the resulting field pattern, look the same if we rotatethem about the line. The second symmetry occurs because the lineis infinite: if we slide the line along its own length, nothing changes.This sliding symmetry, known as a translation symmetry, tells usthat the field must point directly away from the line at any givenpoint.Based on these symmetries, we choose the Gaussian surfaceshown in figure i. If we want to know the field at a distance Rfrom the line, then we choose this surface to have a radius R, as626 Chapter 10 Fields