Symmetries & Idealizations Homework 6 Due 10/17/14 @ 5 pm QUIZ: 1. Calculate the curl of each of the following vector fields. You may look up the formulas for curl in curvilinear coordinates. ~ = z 2 xˆ + x2 yˆ − y 2 zˆ (a) F ~ = e−x xˆ + e−y yˆ + e−z zˆ (b) G ~ = yz xˆ + zx yˆ + xy zˆ (c) H (d) ~I = x2 xˆ + z 2 yˆ + y 2 zˆ ~ = xy xˆ + xz yˆ + yz zˆ (e) J ~ = s2 sˆ (f) K ~ = r3 φˆ (g) L PRACTICE: 2. Choose some simple vector fields of your own and find the curl of them both by hand and using Mathematica or Maple. Choose some that are written in terms of rectangular coordinates and others in cylindrical and/or spherical. 3. If you need more practice visualizing curl, go through the Mathematica Notebook on the course website. 4. A current I flows down a wire of radius a. (a) If it is uniformly distributed over the surface, give a formula for the surface current ~ density K. ~ is inversely (b) If it is distributed in such a way that the volume current density, |J|, ~ proportional to the distance from the axis, give a formula for J. REQUIRED: 5. In this problem, you will be investigating, from several different points of view, a cylindrical wire of finite thickness R, carrying a non-uniform current density J = κr, where κ is a constant and r is the distance from the axis of the cylinder. (a) Find the total current flowing through the wire. 1 (b) Find the current flowing through Disk 2, a central (circular cross-section) portion of the wire out to a radius r2 < R. (c) Use Amp`ere’s law in integral form to find the magnetic field at a distance r1 outside the wire. (d) Use Amp`ere’s law in integral form to find the magnetic field at a distance r2 inside the wire. (e) Use theta functions to write the magnetic field everywhere (both inside and outside of the wire) as a single function. (f) Evaluate Z ~ ~ ~ ∇ × B · dA for Disk 2, a circular disk of radius r2 < R. Use this result and part (d) to verify Stokes’ theorem on this surface. (g) Evaluate Z ~ · dA ~ ~ ×B ∇ for Disk 1, a circular disk of radius r1 > R. Use this result and part c) to verify Stokes’ theorem on this surface. 6. Find the volume current density that produces the following magnetic field (expressed in cylindrical coordinates): 2  µ0 I r ˆ  φ r≤a  2πa2    ~ r) = µ0 I φˆ a ≥ r < b B(~ 2πr      0 r>b What is a physical situation that corresponds to this current density? 7. The gravitational field due to a spherical shell of mass is given by:  0 r<b      b3 4 πρ G r − rˆ b < r < a − ~g = 3 r2   3 3   4  − 3 πρ G a r−b rˆ a < r 2 where b is the inside radius of the shell, a is the outside radius of the shell, and ρ is the constant mass density. (a) Using an explicit line integral, calculate the work required to bring a test mass, of mass m0 , from infinity to a point P , which is a distance c (where c > a) from the center of the shell. (b) Using an explicit line integral, calculate the work required to bring the test mass along the same path, from infinity to the point Q a distance d (where b < d < a) from the center of the shell. (c) Using an explicit line integral, calculate the work required to bring the test mass along the same radial path from infinity all the way to the center of the shell. (d) Using an explicit line integral, calculate the work required to bring in the test mass along the path drawn below, to the point P of question a. Compare the work to your answer from question a. 3 (e) What is the work required to bring the test mass from infinity along the path drawn below to the point P of question a. Explain your reasoning. 4