# Hwk1_additional_solutions - Homework 1 1 The Maxwell...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Homework 1 1. The Maxwell relationship between a time-independent 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 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 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 non-zero 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....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

Hwk1_additional_solutions - Homework 1 1 The Maxwell...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online