Simple Nature - Light and Matter

to a flagpole, we can cancel out a segment of the wire’s currentand create a detour. Ampère’s law is still true because, as shownin the last step, the square dipole makes zero contribution to thecirculation. We can make as many detours as we like in this manner,thereby morphing the wire into an arbitrary shape like f/7.What about a wire like f/8? It doesn’t pierce the Ampèrian surface,so it doesn’t add anything to I through , and we need to show thatit likewise doesn’t change the circulation. This wire, however, couldbe built by tiling the half-plane on its right with square dipoles, andwe’ve already established that the field of a distant dipole doesn’tcontribute to the circulation. (Note that we couldn’t have done thiswith a wire like f/7, because some of the dipoles would have beenright on top of the Ampèrian surface.)If Ampère’s law holds for cases like f/7 and f/8, then it holdsfor any complex bundle of wires, including some that pass throughthe Ampèrian surface and some that don’t. But we can build justabout any static current distribution we like using such a bundle ofwires, so it follows that Ampère’s law is valid for any static currentdistribution.11.3.3 Maxwell’s equations for static fieldsStatic electric fields don’t curl the way magnetic fields do, so wecan state a version of Ampère’s law for electric fields, which is thatthe circulation of the electric field is zero. Summarizing what weknow so far about static fields, we haveΦ E = 4πkq inΦ B = 0Γ E = 0Γ B = 4πkc 2 I through .This set of equations is the static case of the more general relationsknown as Maxwell’s equations. On the left side of each equation, wehave information about a field. On the right is information aboutthe field’s sources.It is vitally important to realize that these equations are onlytrue for statics. They are incorrect if the distribution of chargesor currents is changing over time. For example, we saw on page598 that the changing magnetic field in an inductor gives rise toan electric field. Such an effect is completely inconsistent with thestatic version of Maxwell’s equations; the equations don’t even referto time, so if the magnetic field is changing over time, they willnot do anything special. The extension of Maxwell’s equations tononstatic fields is discussed in section 11.6.676 Chapter 11 Electromagnetism