A Student's Guide to Maxwell's Equations
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Gauss's law for electric fields, Gauss's law for magnetic fields, Faraday's law, and the Ampere-Maxwell law are four of the most influential equations in science. In this guide for students, each equation is the subject of an entire chapter, with detailed, plain-language explanations of the physical meaning of each symbol in the equation, for both the integral and differential forms. The final chapter shows how Maxwell's equations may be combined to produce the wave equation, the basis for the electromagnetic theory of light. This book is a wonderful resource for undergraduate and graduate courses in electromagnetism and electromagnetics. A website hosted by the author at www.cambridge.org/9780521701471 contains interactive solutions to every problem in the text as well as audio podcasts to walk students through each chapter.
induced emf. Example 3.1: Given an expression for the magnetic field as a function of time, determine the emf induced in a loop of specified size.Problem: For a magnetic field given by Find the emf induced in a square loop of side L lying in the xy-plane with one corner at the origin. Also, find the direction of current flow in the loop. Solution: Using Faraday’s flux rule, For a loop in the xy-plane, and da = dx dy, so and Taking the time derivative gives Since upward magnetic flux
you followed a random path around the wire getting closer and farther from the wire as you went around, you’d find the magnetic field getting stronger and weaker, and no longer pointing along your path. Figure 4.1 Exploring the magnetic field around a current-carrying wire. Now imagine keeping track of the magnitude and direction of the magnetic field as you move around the wire in tiny increments. If, at each incremental step, you found the component of the magnetic field along that portion
Figure 4.4(c). An important concept for you to understand is that the enclosed current is exactly the same irrespective of the shape of the surface you choose, provided that the path of integration is a boundary (edge) of that surface. The surfaces shown in Figure 4.4 are the simplest, but you could equally well have chosen the surfaces shown in Figure 4.5, and the enclosed currents would be exactly the same. Figure 4.5 Alternative surfaces with boundaries C1, C2, and C3. Notice that in Figure
wire exactly compensates for the curvature of the field lines. Thus, the curl of the magnetic field is zero everywhere except at the wire itself, where electric current is flowing. The electric current density The right side of the differential form of the Ampere–Maxwell law contains two source terms for the circulating magnetic field; the first involves the vector electric current density. This is sometimes called the “volume current density,” which can be a source of confusion if you’re
127 usefulness of 127 displacement current 94 units of 107 divergence 32–35 in Cartesian coordinates 33 locating regions of 32 main idea 120 in non-Cartesian coordinates 35 relationship to flux 32 divergence theorem 114–115 main idea 114 dot product 6 how to compute 6 physical significance 6 electric field definition of 3 electrostatic vs. induced 1, 62–63 equations for simple objects 5 induced 62–63 direction of 63 units of 3, 62 electromotive force (emf) 68 units of 68