By Daniel Fleisch

Gauss's legislations for electrical fields, Gauss's legislation for magnetic fields, Faraday's legislation, and the Ampere-Maxwell legislation are 4 of the main influential equations in technology. during this advisor for college students, each one equation is the topic of a complete bankruptcy, with distinctive, plain-language factors of the actual that means of every image within the equation, for either the essential and differential kinds. the ultimate bankruptcy indicates how Maxwell's equations can be mixed to provide the wave equation, the foundation for the electromagnetic thought of sunshine. This booklet is a superb source for undergraduate and graduate classes in electromagnetism and electromagnetics. an internet site hosted by way of the writer at www.cambridge.org/9780521701471 comprises interactive suggestions to each challenge within the textual content in addition to audio podcasts to stroll scholars via each one bankruptcy.

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10 Use a special Gaussian surface to prove that the magnitude of the electric field of an infinite flat plane with surface charge density r is j~ Ej ¼ r=2e0 . 11 Find the divergence of the field given by ~ A ¼ ð1=rÞ^r in spherical coordinates. 12 Find the divergence of the field given by ~ A ¼ r^r in spherical coordinates. 13 Given the vector field p^ ~ A ¼ cos py – i þ sinð pxÞ^j; 2 sketch the field lines and find the divergence of the field. 15 Find the charge density in a region for which the electric field in spherical coordinates is given by b cos ðhÞ ^ ^ ~ E ¼ ar 2^r þ h þ cf: r 2 Gauss’s law for magnetic fields Gauss’s law for magnetic fields is similar in form but different in content from Gauss’s law for electric fields.

Irrespective of the shape and location of the closed surfaces placed in the field, all field lines entering the enclosed volume are offset by an equal number of field lines leaving that volume. The physical reasoning behind Gauss’s law should now be clear: the net magnetic flux passing through any closed surface must be zero because magnetic field lines always form complete loops. The next section shows you how to use this principle to solve problems involving closed surfaces and the magnetic field.

To find the electric field outside the sphere, construct your Gaussian surface with radius r > a so that the entire charged sphere is within the Gaussian surface. This means that the enclosed charge is just the charge density times the entire volume of the charged sphere: qenc ¼ ð4=3Þpa3 q. Thus, E¼ ð4=3Þpa3 q qa3 ¼ ðoutside sphereÞ: 2 4pe0 r 3e0 r 2 To find the electric field within the charged sphere, construct your Gaussian surface with r < a. In this case, the enclosed charge is the charge 28 A student’s guide to Maxwell’s Equations density times the volume of your Gaussian surface: qenc ¼ ð4=3Þpr 3 q.