F07_hw07 - Physics 505 Homework Assignment#7 Due Thursday...

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Physics 505 Fall 2007 Homework Assignment #7 — Due Thursday, November 1 Textbook problems: Ch. 5: 5.4, 5.7, 5.8, 5.9 5.4 A magnetic induction ~ B in a current-free region in a uniform medium is cylindrically symmetric with components B z ( ρ, z ) and B ρ ( ρ, z ) and with a known B z (0 , z ) on the axis of symmetry. The magnitude of the axial field varies slowly in z . a ) Show that near the axis the axial and radial components of magnetic induction are approximately B z ( ρ, z ) B z (0 , z ) - ρ 2 4 2 B z (0 , z ) ∂z 2 + · · · B ρ ( ρ, z ) ≈ - ρ 2 ∂B z (0 , z ) ∂z + ρ 3 16 3 B z (0 , z ) ∂z 3 + · · · b ) What are the magnitudes of the neglected terms, or equivalently what is the criterion defining “near” the axis? 5.7 A compact circular coil of radius a , carrying a current I (perhaps N turns, each with current I/N ), lies in the x - y plane with its center at the origin. a ) By elementary means [Eq. (5.4)] find the magnetic induction at any point on the z axis b ) An identical coil with the same magnitude and sense of the current is located
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