P30_045 - s 1 and s 2 ). The connecting paths (each of size...

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45. Consider a circle of radius r , inside the toroid and concentric with it (like either of the loops drawn in Fig. 30-20). The current that passes through the region between this circle and another larger radius circle (well outside the toroid) is Ni ,whe re N is the number of turns and i is the current (note that this region includes a “slice” of the outer rim of the toroid). The current per unit length (of the circle) is λ = Ni/ 2 πr ,and µ 0 λ is therefore µ 0 Ni/ 2 πr , the magnitude of the magnetic ±eld at the circle (call it B 1 ). Since the ±eld outside a toroid (call it B 2 ) is zero, the above result is also the change in the magnitude of the ±eld encountered as you move from the circle to the outside (say, to the larger radius circle mentioned above). The equality is not really surprising in light of Ampere’s law, particularly if the path used in H ~ B · d~s is made to connect the circle in the toroid and the larger radius circle (or
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Unformatted text preview: s 1 and s 2 ). The connecting paths (each of size r ) between the circles can be made perpendicular to the magnetic eld lines (so that ~ B ~s = 0). In fact, we can keep the connecting paths roughly perpendicular to ~ B and manage to have s 1 s 2 if our Amperian loop is very small (especially if r is much smaller than the outer radius of the toroid). Simplifying our notation, the current through the loop is therefore s , so Amperes law yields ( B 1 B 2 ) s = s and B 2 B 1 = . What this demonstrates is that the change of the magnetic eld is when moving from one point to another (in a direction perpendicular to the eld) across a current sheet (as the term is used in problem 39); this principle is useful in any discussion of boundary conditions in electrodynamics applications....
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This note was uploaded on 11/12/2011 for the course PHYS 2001 taught by Professor Sprunger during the Fall '08 term at LSU.

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