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Unformatted text preview: Homework 1 1. The Maxwell relationship between a time-independent magnetic field B and the current density J producing it is . = B J A long cylinder of conducting ionized gas (plasma) occupies the region a < (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 ) for a > and ( ) B B = for a < 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 = (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 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 . (a) Given ( ) , B = B z we have ( ) for 0 for . B C a a = - = < = > B Hence for , a > we get B constant (= B ) and for , a < we get ( ) 1 , B B C =- where B 1 is a constant of integration. Continuity at a = gives 1 . B B Ca = + Applying the condition B = 0 at 0, = we get ( ) ( ) for a for a....
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