Transcript Week 12
Week 12 Vector Potential, magnetic dipoles MD12-1 A toroid has average radius R, winding diameter DR , a total of N windings with current I. We "idealize" this as a surface current running around the surface. What is K? A) I/R C) NI/R B) I/(2 R) D) NI/(2 R) E) Something else MD12-2 A torus has N windings and current I. Answer using Ampere's Law : The magnitude of the magnetic field inside a torus is A) constant, independent of position within the torus B) non-constant, depends on position within torus 5.19 The vector potential in a certain region is given by A(x, y) C y xˆ (C is a positive constant) Consider the imaginary loop shown. What can you say about the magnetic field in this region? A. B. C. D. B is zero B is non-zero, parallel to z-axis B is non-zero, parallel to y-axis B is non-zero, parallel to x-axis y A x 5.25 A 0J 2 In Cartesian coordinates, this means: 2 A J , etc. x 0 x Does it also mean, in spherical coordinates, that 2 ? Ar 0 J r A) Yes B) No 5.25b 0 A (r ) 4 J(r') d ' Can you calculate that integral using spherical coordinates? A) Yes, no problem B) Yes, r' can be in spherical, but J still needs to be in Cartesian components C) No. MD12-3 z The vector potential A due to a long straight wire with current I along the z-axis is in the direction parallel to: I A) zˆ B) ˆ (azimuthal) C) sˆ (radial) A=? MD12-4a,b A circular wire carries current I in the xy plane. What can you say about the vector potential A at the points shown? At point a, the vector potential A is: A) Zero B) Parallel to x-axis C) Parallel to y-axis D) Parallel to z-axis z b a y I x At point b, the vector potential A is: A) Zero B) Parallel to x-axis C) Parallel to y-axis D) Parallel to z-axis E-field around electric dipole B-field around magnetic dipole (current loop) From Purcell, Electricity and Magnetism MD12-5 Two magnetic dipoles m1 and m2 are oriented in three different ways. m1 1. 2. 3. m2 Which ways produce a dipole field at large distances? A) None of these B) All three C) 1 only D) 1 and 2 only E) 1 and 3 only 5.29 This is the formula from Griffiths for a magnetic dipole at the origin is: ˆ rˆ 0 m A (r) 2 4 r Is this the exact vector potential for a flat ring of current with m=Ia, or is it approximate? A) It's exact B) It's exact if |r| > radius of the ring C) It's approximate, valid for large r D) It's approximate, valid for small r 5.26 What is A dl ? A) The current density J B) The magnetic field B C) The magnetic flux B D) It's none of the above, but is something simple and concrete E) It has no particular physical interpretation at all MD12-6 In the plane of a magnetic dipole, with magnetic moment m (out), the vector potential A looks like kinda like this with A ~ 1/r2 At point x, which way does curl(B) point? A) Right B) Left C)In D)Out E) Curl is zero x 5.28b In general, which of the following are continuous as you move past a boundary? A) A B) Not all of A, just Aperp C) Not all of A, just A|| D) Nothing is guaranteed to be continuous regarding A 5.27 Suppose A is azimuthal, given by c ˆ A s What can you say about curl(A)? A) curl(A)=0 everywhere B) curl(A) = 0 everywhere except at s=0. C) curl(A) is nonzero everywhere D) ??? MD12-7 The force on a segment of wire L is F I LB A current-carrying wire loop is in a constant magnetic field B = B z_hat as shown. What is the direction of the torque on the loop? A) Zero B) +x C) +y D) +z E) None of these z I(out) B B y x I(in) z I y 6.1 Griffiths argues that the torque on a magnetic dipole in a B field is: m B How will a small current loop line up if the B field points uniformly up the page? A) m m C) D) B) m m 6.2 Griffiths argues that the force on a magnetic dipole in a B field is: F (m B) If the dipole m points in the z direction, what can you say about B if I tell you the force is in the x direction? A) B simply points in the x direction B) Bz must depend on x C) Bz must depend on z D) Bx must depend on x E) Bx must depend on z