Read Chapters 7 and 11.
1. a) Problem 7.5.I. First, correct the errors in
the equations. All v's should be the components perpendicular to the
shock front, and the vector in 7.10b should be a scalar.
b) In the strong shock limit, determine the pressure P2. Find the upstream and downstream speeds in terms of the local sound speeds c1 and c2 and comment.
2. Magnetic fields can always be derived from a vector potential
and can sometimes be derived from a scalar potential. In this problem,
you'll explore the conditions under which the latter is possible.
a) Let A = MBsin(θ)r-2 &phi where &phi is a unit vector in the azimuthal direction, MB is the magnetic moment, and B = ∇ x A defines the magnetic vector potential A. Find the magnetic field and compare to expressions in the text for a magnetic dipole.
b) Show that ∇ ⋅ B = 0 for all possible A.
c) Given B = ∇φ, evaluate ∇ x B and ∇ ⋅ B and give condition under which each is zero.
d) Compare your answers to c) to Maxwell's equations and comment. Are your expressions valid deep inside the planet?
3. a) Derive Eq. 7.16 from Maxwell's Equations.
b) Derive Equation 7.18 from Eq. 7.16. Determine the cgs units of the electrical conductivity σo.
4. Problem 11.4.E.
5. Problem 11.12.I.