Week 12

Vector Potential, magnetic dipoles

A toroid has average radius R, winding diameter R , 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 B) I/(2 R) C) NI/R D) NI/(2 R)

E) Something else

MD12-1

The magnitude of the magnetic field inside a torus is

A)constant, independent of position within the torusB) non-constant, depends on position within torus

A torus has N windings and current I.

Answer using Ampere's Law :

MD12-2

The vector potential in a certain region is given by

(C is a positive constant) Consider the imaginary loop shown. What can you say about the magnetic field in this region?

A. B is zeroB. B is non-zero, parallel to z-axisC. B is non-zero, parallel to y-axisD. B is non-zero, parallel to x-axis

ˆA(x, y) C y x

5.19

x

y

A

2A 0

J

In Cartesian coordinates, this means:, etc.

Does it also mean, in spherical coordinates, that

?

2Ax 0Jx

2Ar 0J r

A) YesB) No

5.25

A (

r )

0

4

J (r')

d '

Can you calculate that integral using spherical coordinates?

A) Yes, no problemB) Yes, r' can be in spherical, but J still needs to be in

Cartesian components

C) No.

5.25b

The vector potential A due to a long straight wire with current I along the z-axis is in the direction parallel to:

z

I A = ?

ˆA) z

ˆB) (azimuthal)

ˆC) s (radial)

MD12-3

A circular wire carries current I in the xy plane. What can you say about the vector potential A at the points shown?

x

y

z

a

b

IAt point b, the vector potential A is:A) ZeroB) Parallel to x-axisC) Parallel to y-axisD) Parallel to z-axis

At point a, the vector potential A is:A) ZeroB) Parallel to x-axisC) Parallel to y-axisD) Parallel to z-axis

MD12-4a,b

From Purcell,Electricity and Magnetism

E-field aroundelectric dipole

B-field aroundmagnetic dipole(current loop)

Two magnetic dipoles m1 and m2 are oriented in three different ways.

m1 m2

1.

2.

3.

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

MD12-5

This is the formula from Griffiths for a magnetic dipole at the origin is:

02

ˆ ˆm r(r)

4 r

A

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

Is this the exact vector potential for a flat ring of current with m=Ia, or is it approximate?

5.29

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

5.26

dlA

What is ?

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

x

At point x, which way does curl(B) point?A)Right B)LeftC)InD)OutE)Curl is zero

MD12-6

In general, which of the following are continuous as you move past a boundary?

5.28b

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

Suppose A is azimuthal, given by

What can you say about curl(A)?

A

c

sˆ

A) curl(A)=0 everywhere

B) curl(A) = 0 everywhere except at s=0.

C) curl(A) is nonzero everywhere

D) ???

5.27

F I L B

The force on a segment of wire L is

B

y

z I(in)

I(out)

x

y

z

B

I

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) +zE) None of these

MD12-7

Griffiths argues that the torque on a magnetic dipole in a B field is:

m

B

6.1

How will a small current loop line up if the B field points uniformly up the page?

A)

B)

C) D)

m

m

m m

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 directionB) Bz must depend on xC) Bz must depend on zD) Bx must depend on xE) Bx must depend on z

6.2