Chapt. 31Faraday’s Law of Induction and Inductors239evaluate it in terms ofB,ω,θ, the cylindrical radial coordinater, and the differentialdz.Note that there are two cases. The differential bit of contour on the side of the contourcounterclockwise from ˆnhas velocity−rωˆnand that on the side of the contour clockwisefrom ˆnhas velocity in the directionrωˆn.HINT:Note that ˆz·dvectors=dzwheredzcould beimplicitly positive for negative.c) To absorb the two cases for the two sides of ˆnin a single formula let’s introduce coordinateywhich is in the plane of the contour and perpendicular to thezaxis. We lety>0 for thecounterclockwise side andy <0 for the clockwise side. We can now right±rasy. Givethe formula for the differential motional emf now.d) We want to integrate the differential emf over the contour by summing up the contributionsto eachdzinterval first about each heightzfirst and then summing these up for the total.
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Magnetic Field, Summation, Clockwise, Law of Induction and Inductors