5.24If the arrows represent observer the vector potential A (note that A is the same everywhere), is there a nonzero B in the dashed region?

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1 QUZ: No quiz on Thursday Static Vector Fields Homework 6 Due 11/8/13 PRACTCE: Here are some great quick problems (little to no calculation) to test your knowledge before the exam. Bring your questions to class on Friday For the following situation what do you expect the direction of the A-field to be like at the observer? 1. (a) For the following situation, what will the observer see for the direction of A? For the direction of B? 5.24f the arrows represent observer the vector potential A (note that A is the same everywhere), is there a nonzero B in the dashed region? (b) The arrows below represent the vector potential A (where A is the same everywhere). s there a non-zero B in the dashed region? f so, what direction does it point? How do you know? A. Yes B. No 2. Below are two sketches of electric field lines. For this problem, explanations will count for the majority of the points. C.Need more information to decide 1

2 Which of the following two fields has zero divergence? (a) Which of the do above sketches show field linesonly that violateone of zero Maxwell s EquaA) Both B) is tions within the region bounded by the dashed lines? How do you know? (b) For those that do not Maxwell s Equations, what current be needed C) Only isviolate zero D) Neither iswould zero to generate the field and where would it be located? E)?? ~ at the point S? 3. What is B s 4. An electron is moving in a straight line with constant speed v. What method would you use to calculate the B-field generated by this electron? v e5. Which of the below B-field components are not possible? How do you know? 2

3 A "ribbon" (width a) of surface current flows (with surface current 6. You have a charge distribution composed of two point charges: one with charge +3q density located K) at x = d and the other with charge q located at x = +d. Right (a) next Sketchto the it charge is distribution. a second identical (b) Write an expression for the volume charge density ρ( r) everywhere in space. ribbon of current. 7. Sketch the volume charge density: ρ(x, y, z) = c δ(x 3) Viewed collectively, what is the new total surface current density? 8. A ribbon (width a) of surface current flows with surface current density K. Right next to it is a second identical ribbon of current. A) K B) 2K C) K/2 D) Something else a (a) Viewed collectively, what is the new total surface current density? (b) Viewed collectively, what is the new total current? 9. Two charged rings of radius R spin in opposite directions, each with total current. They are placed a distance 2L apart and oriented as shown below. 3

4 z Loop 2 R 2L R Loop 1 (a) What is the magnetic field on the z-axis due to Loop 1? (b) What is the magnetic field on the z-axis due to Loop 2? (c) What is the leading non-zero term for the total magnetic field on the z-axis near the midpoint between the coils (z << R)? REQURED: 10. Find the volume current density that produces the following magnetic field (expressed in cylindrical coordinates): B( r) = µ 0 r 2πa 2 ˆφ µ 0 2πr ˆφ r a a r < b 0 r > b What is a physical situation that corresponds to this current density? 11. Consider a point a distance z above the center of an infinitesimally thin, square sheet of current. The current is parallel to one of the square sides. (Obviously, since the current cannot just begin and end in the middle of nowhere, this current is just the building block for some larger current.) (a) Use the Biot-Savart Law to find the magnetic field at the point z. You may use any symmetry arguments you like, but do not use Ampere s Law. Note: if you choose to use Mathematica or Maple to evaluate the integral, it may take you into complex number land, even though the integral is clearly real. To address this issue, you should be explicit about what assumptions you want the program to make ( Assume in Maple and Assumptions in Mathematica) 4

5 (b) Consider your previous answer in the limit that the square becomes infinitely large. (c) Discuss your answer in the light of the magnetic field above an infinite sheet of current as found using Ampere s Law. 12. n class we derived an expression, using the Biot-Savart law, for the magnetic field created by a rotating ring of charge (total charge Q, radius R, rotating with period T ) everywhere in space. B( r) = µ 0 Q R 2π 4π T 0 z cos φ ˆr + (R r cos φ ) ẑ (r 2 + R 2 2rR cos φ + z 2 ) 3/2 dφ (a) Take the limit r >> R in the plane of the ring, and simplify to find the leading non-zero term. Briefly describe any checks you did to validate your answer. 13. n 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. (b) Find the current flowing through Disk 2, a central (circular cross-section) portion of the wire out to a radius r 2 < R. (c) Use Ampère s law in integral form to find the magnetic field at a distance r 1 outside the wire. 5

6 (d) Use Ampère s law in integral form to find the magnetic field at a distance r 2 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 ( ) B da for Disk 2, a circular disk of radius r 2 < R. Use this result and part (d) to verify Stokes theorem on this surface. (g) Evaluate ( ) B da for Disk 1, a circular disk of radius r 1 > R. Use this result and part c) to verify Stokes theorem on this surface. 6