Hwk1_additional - of a stream function by 2 1 1 , . sin sin...

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Homework 1 1. The Maxwell relationship between a time-independent magnetic field B and the current density J producing it is 0 . μ ∇× = B J A long cylinder of conducting ionized gas (plasma) occupies the region a ρ < (in cylindrical polar co-ordinates. (a) Show that a uniform current density (0, C , 0) and a magnetic field (0, 0, B ) with B = constant (= B 0 ) for a > and ( ) B B = for a < are consistent with this Maxwell relationship. Obtain expressions for C and B in terms of B 0 and a , given that the magnetic field is continuous at . a = (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 non-zero component, ( ) . A φ Find ( ) , A which is also continuous at . a = (c) The gas pressure ( ) p satisfies the hydrostatic equation p ∇ = × J B and vanishes at the outer wall of the cylinder. Find p . 2. In a viscous flow, the velocity components in spherical polar co-ordinates are given in terms
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Unformatted text preview: of a stream function by 2 1 1 , . sin sin r u u r r r = = - (a) Find an explicit expression for ( ) sin . E r = - u ( E is to be considered as a differential operator.) (b) The stream function satisfies 2 E = and, for the flow past a sphere, ( ) ( ) 2 , sin . r f r = Show that ( ) 4 4 2 4 8 8 0. r f r f rf f -+-= 3. A vector field is defined by cos cos cos sin sin . x z y z z F z F z a a a = + + = + F i j k e k Use the divergence theorem to calculate the flux of F through the closed surface bounded by the cylinders a = and 2 a = and the planes 2. z a =...
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This note was uploaded on 12/02/2011 for the course PHYS 809 taught by Professor Macdonald during the Fall '11 term at University of Delaware.

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