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Unformatted text preview: Homework 1 1. The Maxwell relationship between a timeindependent magnetic field B and the current density J producing it is . μ ∇× = B J A long cylinder of conducting ionized gas (plasma) occupies the region a ρ < (in cylindrical polar coordinates). (a) Show that a uniform current density (0, C , 0) and a magnetic field (0, 0, B ) with B = constant (= B ) for a ρ > and ( ) B B ρ = for a ρ < are consistent with this Maxwell relationship. Obtain expressions for C and B in terms of B and a , given that the magnetic field is continuous at . a ρ = (b) The magnetic field can be expressed as , = ∇× B A where A is the vector potential. Show that a suitable A can be found with only one nonzero component, ( ) . A φ ρ Find ( ) , A φ ρ which is also continuous at . a ρ = (c) The gas pressure ( ) p ρ satisfies the hydrostatic equation p ∇ = × J B and vanishes at the outer wall of the cylinder. Find p . (a) Given ( ) ˆ, B ρ = B z we have ( ) ˆ ˆ for 0 for . B C a a ρ μ ρ ρ ρ ∂ ∇× =  = < ∂ = > B φ φ Hence for , a ρ > we get B constant (= B ) and for , a ρ < we get ( ) 1 , B B C ρ μ ρ = where B 1 is a constant of integration. Continuity at a ρ = gives 1 . B B Ca μ = + Applying the condition B = 0 at 0, ρ = we get ( ) ( ) for a for a....
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 Fall '11
 MacDonald
 Current, Work, Magnetic Field, Sin, aφ

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