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hwk1 - Y = y-v x Ω with Ω = qB/mc Hint the arguments of f...

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Physics 762 Homework #1 Spring ‘11 Dr. Drake 1. The entropy of a distribution of particles f(v,x) with charge q is given by S = - Z dx Z dvf ( x, v ) ln f ( x, v ) . (1) Show that by minimizing S subject to the constraint that the total number of particles R dx R dvf and the total energy R dx R dv ( mv 2 / 2 + ) f are conserved that the equilibrium distribution is a Maxwellian with temperature T and den- sity n ( x ). Assume that the potential φ ( x ) is given. Hint: Use Lagrange multipliers when you minimize S to ensure the total particle number and energy are conserved. 2. Show that f ( v z , v 2 x + v 2 y , X, Y ) is a solution of the Vlasov equation in a uniform magnetic field B = B ˆ z , where X = x + v
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Unformatted text preview: Y = y-v x / Ω with Ω = qB/mc . Hint: the arguments of f are all constants of motion. What are X and Y ? 3. A cylindrically symmetric plasma has a distribution function f given in terms of the energy H and canonical angular momentum P θ = r ( mv θ + qA θ /c ), which are the particle constants of motion, f ( r, v ) = n (2 πT/m ) 3 / 2 exp (-H-λP θ T ) , (2) where n , λ and T are constants. The vector potential A θ ( r ) produces a uniform magnetic field. Find expressions for the density and rotation rate as a function of r ....
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