Simple Nature - Light and Matter

Note that the curl, just like a derivative, has a differential dividedby another differential. In terms of this definition, we find Ampère’slaw in differential form:curl B = 4πkc 2jb / The coordinate systemused in the following examples.The complete set of Maxwell’s equations in differential form is collectedon page 914.11.4.2 Properties of the curl operatorThe curl is a derivative.As an example, let’s calculate the curl of the field ˆx shown infigure c. For our present purposes, it doesn’t really matter whetherthis is an electric or a magnetic field; we’re just getting out feet wetwith the curl as a mathematical definition. Applying the definitionof the curl directly, we construct an Ampèrian surface in the shapeof an infinitesimally small square. Actually, since the field is uniform,it doesn’t even matter very much whether we make the squarefinite or infinitesimal. The right and left edges don’t contribute tothe circulation, since the field is perpendicular to these edges. Thetop and bottom do contribute, but the top’s contribution is clockwise,i.e., into the page according to the right-hand rule, while thebottom contributes an equal amount in the counterclockwise direction,which corresponds to an out-of-the-page contribution to thecurl. They cancel, and the circulation is zero. We could also havedetermined this by imagining a curl-meter inserted in this field: thetorques on it would have canceled out.It makes sense that the curl of a constant field is zero, becausethe curl is a kind of derivative. The derivative of a constant is zero.The curl is rotationally invariant.Figure c looks just like figure c, but rotated by 90 degrees. Physically,we could be viewing the same field from a point of view thatwas rotated. Since the laws of physics are the same regardless ofrotation, the curl must be zero here as well. In other words, the curlis rotationally invariant. If a certain field has a certain curl vector,then viewed from some other angle, we simply see the same field andthe same curl vector, viewed from a different angle. A zero vectorviewed from a different angle is still a zero vector.As a less trivial example, let’s compute the curl of the field F =xŷ shown in figure e, at the point (x = 0, y = 0). The circulationaround a square of side s centered on the origin can be approximatedc / The field ˆx.d / The field ŷ.e / The field xŷ.Section 11.4 Ampère’s Law In Differential Form (Optional) 679