P31_031 - 31(a Letting x be the distance from the right end...

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31. (a) Letting x be the distance from the right end of the rails to the rod, we Fnd an expression for the magnetic flux through the area enclosed by the rod and rails. By Eq. 30-19, the Feld is B = µ 0 i/ 2 πr , where r is the distance from the long straight wire. We consider an inFnitesimal horizontal strip of length x and width dr , parallel to the wire and a distance r from it; it has area A = xdr and the flux d Φ B =( µ 0 ix/ 2 ) dr . By Eq. 31-3, the total flux through the area enclosed by the rod and rails is Φ B = µ 0 ix 2 π Z a + L a dr r = µ 0 ix 2 π ln µ a + L a . According to ±araday’s law the emf induced in the loop is E = d Φ B dt = µ 0 i 2 π dx dt ln µ a + L a = µ 0 iv 2 π ln µ a + L a = (4 π × 10 7 T · m / A)(100A)(5 . 00 m / s) 2 π ln µ 1 . 00 cm + 10 . 0cm 1 . 00 cm =2 . 40 × 10 4 V . (b) By Ohm’s law, the induced current is i ` = E /R =(2 . 40 × 10 4 V) / (0 . 400 Ω) = 6 . 00 × 10 4 A. Since the flux is increasing the magnetic Feld produced by the induced current must be into the page in the region enclosed by the rod and rails. This means the current is clockwise.
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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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