edyn - Physics 54 Time-Varying Fields The only cure for...

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Unformatted text preview: Physics 54 Time-Varying Fields The only cure for boredom is curiosity. There is no cure for curiosity. Dorothy Parker Completion of Amperes Law As an example of a time-varying situation, consider the B-Feld established by the current charging a capacitor. Shown is a parallel-plate capacitor with circular plates. We are interested in the B-Feld at point P , shown in the edge-on view. This situation has axial symmetry, which we exploit to use Ampere's law and calculate B . The Feld lines are circles around the symmetry axis, so we choose as our path of integration a circle through point P . We Fnd B d r = 2 rB = I linked . Here I linked is the current passing through the area bounded by the circular path. But what area? The simplest is a circular disk bounded by the circle of radius r . Because there is no current through such a disk, we would have I linked = , from which we predict B = at P . But experimentally the Feld is not zero at P . Ampere's law gives a prediction that is wrong. However, we could use a different surface bounded by the circular path. Consider a bowl-shaped surface with the circle as its rim; the bowl surface can be chosen to go around the left plate so that the wire penetrates through it. In that case we would conclude that I linked = I , leading to B = I 2 r . This is in agreement with experiment. These different choices of surface bounded by the path give contradictory results. How do we know which surface to use? Amperes law itself gives no instruction on this point. No law of nature can be this vague. It must be either modiFed or discarded. I a I P r PHY 54 1 Electrodynamics Now consider the fux oF the E-eld through these two surFaces bounded by the circle. irst use the circular disk surFace. Outside the region between the plates the E-eld is negligible, so the fux oF E through the disk surFace is that through a circle oF radius a: e = E d A = a 2 E = a 2 Q a 2 = Q . (Here we have used E = / and = Q / a 2 .) rom this we nd d e dt = dQ dt = I . This shows that the quantity d e / dt has the dimensions oF a current, which in this case is the same as the current in the wires. Maxwell called it the displacement current . The term displacement arose from a model Maxwell had in mind, and has no signiFcance today. He suggested that Amperes law needed to be completed by adding the displacement current to the true current, so that the law reads B d r = I linked + d dt E d A . Now let us apply this to our capacitor situation to see how it resolves the ambiguity. IF we choose the disk surFace, then I linked = but d dt E d A = I , to we nd B = I 2 r , as we should. IF we choose the bowl surFace we have I linked = I and d dt E d A = , again leading to B = I 2 r . The choice oF surFace no longer matters....
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This note was uploaded on 10/19/2011 for the course PHYSICS 54L taught by Professor Thomas during the Summer '09 term at Duke.

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edyn - Physics 54 Time-Varying Fields The only cure for...

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