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{\bf Introduction to Astrophysics II, 29:120\\
     Winter 2009\\
First Homework Set... January 22, 2009. Due January 29, 2009}\\
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Show calculations and give reasons for your answers. Don't go around 
confused and despondent; if you do not know how to get started, ask me for help. The purpose
of problem sets is to promote thinking and lead to understanding, not 
produce a {\em confiteor} of revealed truth.\\


\noindent
{\bf (1)} A hydrogen plasma has a number density of $5 \times 10^6$ atoms/m$^3$, and there is a magnetic field $B_0=5 \times 10^{-9}$ Tesla.  What is the Alfv\'{e}n speed in this plasma? \\

\noindent
{\bf (2)} Show by direct substitution that waves are solutions to the equation derived in class, 
\begin{equation}
\frac{\partial^2 b}{\partial t^2} - V_A^2 \frac{\partial^2 b}{\partial z^2} = 0 
\end{equation} \\

\noindent
{\bf (3)} (a) Write down the wave function $b(z,t)$ corresponding to the magnetic field in an Alfv\'{e}n Wave.  (b) Using intermediate equations derive in class, derive  an expression for the velocity wave function $v_x(z,t)$. \\

\noindent
{\bf (4)} An Alfv\'{e}n wave is propagating in the plasma described in problem \# 1.  It has an amplitude of $|b_x| = 0.1 B_0$ (as defined in lecture) and a wavelength of $\lambda = 7 \times 10^5$ km.  What is the maximum acceleration of the plasma due to the presence of this wave?\\

\noindent
{\bf (5)} If the Alfv\'{e}n wave described in Problem \# 4 were propagating in a single direction, what would the associated energy flux be?  {\bf Hint:} The energy flux $S$ is given by $S= \rho_E V$ where $\rho_E$ is the energy density and $V$ is the speed at which the energy is transported.\\ 

\noindent
{\bf (6)} Consider an imaginary spherical surface of radius $R_0$ in a moving, constant density fluid.  Measurements show that the fluid velocity on the surface of the sphere is given by 
\begin{equation}
\vec{v}(r=R_0,\theta,\phi) = (V_0 \cos \theta + V_1 \sin \theta ) \hat{e}_r - V_0 \sin \theta \hat{e}_{\theta}
\end{equation}
Explain why this information tells you that there is a source of fluid interior to the sphere.  Derive an expression for the rate at which mass is being added to the fluid inside the sphere. {\bf Hint:} Go back and look at the equation of continuity from the equations of hydrodynamics, and think about Gauss's theorem from electricity and magnetism.  
\\

\noindent
{\bf (7)} Problem 12.1\\

\noindent
{\bf (8)} Problem 12.2\\


 
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