By Daniel Fleisch
Gauss's legislation for electrical fields, Gauss's legislations for magnetic fields, Faraday's legislations, and the Ampere-Maxwell legislations are 4 of the main influential equations in technology. during this consultant for college kids, every one equation is the topic of a whole bankruptcy, with unique, plain-language factors of the actual that means of every image within the equation, for either the fundamental and differential varieties. the ultimate bankruptcy indicates how Maxwell's equations should be mixed to provide the wave equation, the root for the electromagnetic conception of sunshine. This booklet is an excellent source for undergraduate and graduate classes in electromagnetism and electromagnetics. an internet site hosted by way of the writer at www.cambridge.org/9780521701471 includes interactive options to each challenge within the textual content in addition to audio podcasts to stroll scholars via each one bankruptcy.
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Extra resources for A Student's Guide to Maxwell's Equations
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. For both electric and magnetic fields, the integral form of Gauss’s law involves the flux of the field over a closed surface, and the differential form specifies the divergence of the field at a point.
It is the ability of dielectric materials to reduce the amplitude of an electric field that leads to their most common application: increasing the capacitance and maximum operating voltage of capacitors. 10 Electric field induced in a dielectric. C¼ eA ; d where A is the plate area, d is the plate separation, and e is the permittivity of the material between the plates. High-permittivity materials can provide increased capacitance without requiring larger plate area or decreased plate spacing. The permittivity of a dielectric is often expressed as the relative permittivity, which is the factor by which the material’s permittivity exceeds that of free space: relative permittivity er ¼ e=e0 : Some texts refer to relative permittivity as ‘‘dielectric constant,’’ although the variation in permittivity with frequency suggests that the word ‘‘constant’’ is better used elsewhere.
Problem: A closed cylinder of height h and radius R is placed in a magnetic field given by ~ B ¼ B0 ð ^j À ^kÞ. If the axis of the cylinder is aligned along the z-axis, find the flux through (a) the top and bottom surfaces of the cylinder and (b) the curved surface of the cylinder. Solution: Gauss’s law tells you that the magnetic flux through the entire surface must be zero, so if you’re able to figure out the flux through some portions of the surface, you can deduce the flux through the other portions.
A Student's Guide to Maxwell's Equations by Daniel Fleisch