Simple Nature - Light and Matter

11.3 Magnetic Fields by Ampère’s Law11.3.1 Ampère’s lawAs discussed at the end of subsection 11.2.4, our goal now is tofind an equation for magnetism that, unlike the Biot-Savart law, willnot end up being a dead end when we try to extend it to nonstaticsituations. 6 Experiments show that Gauss’ law is valid in both staticand nonstatic situations, so it would be reasonable to look for anapproach to magnetism that is similar to the way Gauss’ law dealswith electricity.How can we do this? Figure a, reproduced from page 664, is ourroadmap. Electric fields spread out from charges. Magnetic fieldscurl around currents. In figure b/1, we define a Gaussian surface,and we define the flux in terms of the electric field pointing outthrough this surface. In the magnetic case, b/2, we define a surface,called an Ampèrian surface, and we define a quantity called thecirculation, Γ (uppercase Greek gamma), in terms of the magneticfield that points along the edge of the Ampèrian surface, c. Webreak the edge into tiny parts s j , and for each of these parts, wedefine a contribution to the circulation using the dot product of dswith the magnetic field:a / The electric field of a sheet ofcharge, and the magnetic field ofa sheet of current.Γ = ∑ s j · B jThe circulation is a measure of how curly the field is. Like a Gaussiansurface, an Ampèrian surface is purely a mathematical construction.It is not a physical object.In figure b/2, the field is perpendicular to the edges on the ends,but parallel to the top and bottom edges. A dot product is zerowhen the vectors are perpendicular, so only the top and bottomedges contribute to Γ. Let these edges have length s. Since thefield is constant along both of these edges, we don’t actually haveto break them into tiny parts; we can just have s 1 on the top edge,pointing to the left, and s 2 on the bottom edge, pointing to the right.The vector s 1 is in the same direction as the field B 1 , and s 2 is inthe same direction as B 2 , so the dot products are simply equal tothe products of the vectors’ magnitudes. The resulting circulationisΓ = |s 1 ||B 1 | + |s 2 ||B 2 |= 2πkηsc 2+ 2πkηsc 2= 4πkηsc 2 .But ηs is (current/length)(length), i.e., it is the amount of currentthat pierces the Ampèrian surface. We’ll call this current I through .b / A Gaussian surface andan Ampèrian surface.c / The definition of the circulation,Γ.6 If you didn’t read this optional subsection, don’t worry, because the point isthat we need to try a whole new approach anyway.Section 11.3 Magnetic Fields by Ampère’s Law 673