1P14-

Workshop: Using Visualization in Teaching Introductory E&M

AAPT National Summer Meeting, Edmonton, Alberta, Canada.

Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy

P14- 2

MIT Class: Sources of Magnetic Fields

Creating Fields: Biot-SavartExperiment: Magnetic Fields

Ampere’s Law

P14- 3

Magnetic Fields

P14- 4

Gravitational – Electric Fields

2ˆ

mGr

g rMass m Charge q (±)

2ˆ

e

qkr

E r

g mF g

E qF E

Create:

Feel:

Also saw… Dipole p

τ p E

Creates:

Feels:

P14- 5

Magnetism – Bar Magnet

Like poles repel, opposite poles attract

P14- 6

Demonstration:Magnetic Field Lines

from Bar Magnet

P14- 7

Demonstration:Compass (bar magnet) in

Magnetic Field Linesfrom Bar Magnet

P14- 8

Magnetic Field of Bar Magnet

(1) A magnet has two poles, North (N) and South (S)

(2) Magnetic field lines leave from N, end at S

P14- 9

Bar Magnets Are Dipoles!

NO! Magnetic monopoles do not exist in isolation

• Create Dipole Field

• Rotate to orient with Field

Is there magnetic “mass” or magnetic “charge?”

P14- 10

Magnetic Monopoles?

Magnetic monopoles do not exist in isolation

q-qpElectric Dipole

When cut:

2 monopoles (charges)

μ

Magnetic Dipole

When cut: 2 dipoles

Another Maxwell’s Equation! (2 of 4)

0S

inqd

E A

S

0d B A

Gauss’s Law Magnetic Gauss’s Law

P14- 11

PRS:B Field inside a Magnet

P14- 12

PRS: Magnetic Field LinesThe picture shows the field lines outside a permanent magnet The field lines inside the magnet point:

0%

0%

0%

0%

0%

0% 1. Up

2. Down

3. Left to right

4. Right to left

5. The field inside is zero

6. I don’t know 15

P14- 13

PRS Answer: Magnetic Field Lines

Magnetic field lines are continuous.

E field lines begin and end on charges.

There are no magnetic charges (monopoles) so B field lines never begin or end

Answer: 1. They point up inside the magnet

P14- 14

Magnetic Field of the Earth

North magnetic pole located in southern hemisphere

Also a magnetic

dipole!

P14- 15

Fields: Grav., Electric, Magnetic

2ˆ

mGr

g rMass m Charge q (±)

2ˆ

e

qkr

E r

g mF g

E qF E

Create:

Feel:

Dipole p Dipole

τ p E

Create:

Feel: τ μ B

No Magnetic

Monopoles!

E

B

P14- 16

What is B?

B is the magnetic field

It has units of Tesla (T)

N1Tesla 1

A m

This class & next: creating B fields

Next two classes: feeling B fields

P14- 17

How Big is a Tesla?

• Earth’s Field

• Brain (at scalp)

• Refrigerator Magnet

• Inside MRI

• Good NMR Magnet

• Biggest in Lab

• Biggest in Pulsars

5 x 10-5 T = 0.5 Gauss

~1 fT

3 T

18 T

150 T (pulsed)

P14- 18

How do we create fields?

P14- 19

What creates fields?

Magnets – more about this later

The Earth How’s that work?

P14- 20

Magnetic Field of the Earth

North magnetic pole located in southern hemisphere

(for now)

Also a magnetic

dipole!

P14- 21

What creates fields?

Magnets – more about this later

The Earth How’s that work?

Moving charges!

P14- 22

Electric Field Of Point ChargeAn electric charge produces an electric field:

2

1ˆ

4 o

q

rE r

r : unit vector directed from q to P

r

P14- 23

Magnetic Field Of Moving Charge

Moving charge with velocity v produces magnetic field:

2

ˆx

4o q

r

v r

B

:r unit vector directed from q to Pr

permeability of free space7

0 4 10 T m/A

P

P14- 24

Recall:Cross Product

P14- 25

Notation Demonstration

OUT of page

“Arrow Head”

X X

XX

X X

XX

X X

XX

X X

XX

INTO page

“Arrow Tail”

P14- 26

Cross Product: MagnitudeComputing magnitude of cross product A x B:

xC A B

sinC A B

area of parallelogram| C |:

P14- 27

Cross Product: Direction

Right Hand Rule #1:

xC A B

For this method, keep your hand flat!

1) Put Thumb (of right hand) along A

2) Rotate hand so fingers point along B

3) Palm will point along C

P14- 28

Cross Product: Signs

jkijik

ijkikj

kijkji

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

Cross Product is Cyclic (left column)

Reversing A & B changes sign (right column)

P14- 29

PRS Questions:Right Hand Rule

P14- 30

PRS: Cross Product

What is the direction of A x B given the following two vectors?

A B

0%

0%

0%

0%

0%

0%

0% 1. up

2. down

3. left

4. right

5. into page

6. out of page

7. Cross product is zero (so no direction)

15

P14- 31

PRS Answer: Cross Product

Using your right hand, thumb along A, fingers along B, palm into page

Answer: 5. A x B points into the page

A B

P14- 32

PRS: Cross Product

What is the direction of A x B given the following two vectors?

A B

0%

0%

0%

0%

0%

0%

0% 1. up

2. down

3. left

4. right

5. into page

6. out of page

7. Cross product is zero (so no direction)

15

P14- 33

PRS Answer: Cross Product

Using your right hand, thumb along A, fingers along B, palm out of page

Also note from before, one vector flipped so result does too

Answer: 6. A x B points out of the page

A B

P14- 34

Continuous charge distributions:Currents & Biot-Savart

Moving

^

P14- 35

d dqB v

From Charges to Currents?

mcharge

s

d IdB s

chargem

s

v

dq

P14- 36

The Biot-Savart LawCurrent element of length ds carrying current I produces a magnetic field:

20 ˆ

4 r

dI rsBd

(Shockwave)

P14- 37

The Right-Hand Rule #2

ˆ ˆˆ z ρ φ

P14- 38

Animation:Field Generated by a Moving Charge

P14- 39

Demonstration:Field Generated by Wire

P14- 40

Example : Coil of Radius R

Consider a coil with radius R and current I

II

IP

Find the magnetic field B at the center (P)

P14- 41

Example : Coil of Radius R

Consider a coil with radius R and current I

II

IP

1) Think about it:• Legs contribute nothing

I parallel to r• Ring makes field into page

2) Choose a ds3) Pick your coordinates4) Write Biot-Savart

P14- 42

Example : Coil of Radius RIn the circular part of the coil…

ˆd s r

rsd

II

I

02

ˆ

4

dIdB

r

s rBiot-Savart:

ˆ| d | ds s r

024

I ds

r

024

I R d

R

0

4

I d

R

P14- 43

II

I

Example : Coil of Radius RConsider a coil with radius R and current I

sd

0

4

I ddB

R

20

0 4

I dB dB

R

2

0 0

0

24 4

I Id

R R

0 into page2

I

R

B

P14- 44

Example : Coil of Radius R

Notes:•This is an EASY Biot-Savart problem:

• No vectors involved•This is what I would expect on exam

II

IP page into

20

R

IB

P14- 45

PRS Questions:B fields Generated by Currents

P14- 46

PRS: Biot-Savart

The magnetic field at P points towards the

0%

0%

0%

0%

0%

0%

0% 1. +x direction2. +y direction3. +z direction4. -x direction5. -y direction6. -z direction7. Field is zero (so no direction)

1515

P14- 47

PRS Answer: Biot-Savart

The vertical line segment contributes nothing to the field at P (it is parallel to the displacement). The horizontal segment makes a field out of the page.

Answer: 3. B(P) is in the +z direction (out of page)

ikj

P14- 48

PRS: Bent WireThe magnetic field at P is equal to the field of:

0%

0%

0%

0% 1. a semicircle2. a semicircle plus the field of a long straight wire3. a semicircle minus the field of a long straight wire4. none of the above

15

P14- 49

PRS Answer: Bent Wire

All of the wire makes B into the page. The two straight parts, if put together, would make an infinite wire. The semicircle is added to this to get the complete field

Answer: 2. Semicircle + infinite wire

P14- 50

Group Problem:B Field from Coil of Radius R

Consider a coil made of semi-circles of radii R and 2R and carrying a current I

What is B at point P?

P

I

P14- 51

Group Problem:B Field from Coil of Radius R

Consider a coil with radius R and carrying a current I

What is B at point P? WARNING:

This is much harder than the previous problem. Why??

P14- 52

Experiment:Magnetic Fields:Bar Magnets &

Wire Coils

P14- 53

PRS Question:Part I: B Field from Bar Magnet

P14- 54

PRS: Bar Magnet B FieldThinking of your map of the B field lines from part 1, assume that your magnet and compass were on the table in the orientation shown.

N N S

?

The red end of the compass points:

0%

0%

0%

0%

0%

0%

0%

0% 1. Up

2. Down

3. Right

4. Left

5. Up & right

6. Up & left

7. Down & right

8. Down & left 0

P14- 55

PRS Answer: Bar Magnet B Field

If you only had to consider the bar magnet (for example, if you were very close to it) the compass would point to the right. But the Earth’s magnetic field (pointing toward geographic North) pulls the field down.

Answer: 7. Down & right

N N S

P14- 56

Visualization:Bar Magnet &

Earth’s Magnetic Field

P14- 57

PRS Question:Part 3: B Field from Helmholtz

P14- 58

PRS: HelmholtzIdentify the three field profiles that you measured as Single (Sgl),

Helmholtz (Hh) or Anti-Helmholtz (A-H):

-2 -1 0 1 2

0Anti-Helmholtz

Single Coil

Helmholtz

Top CoilBottom Coil

Ma

gn

etic

Fie

ld A

mp

litu

de

Distance along the central axis (z/R)

AB

C

0%0%0%0%0%0% 1. Sgl, Hh, A-H

2. Hh, A-H, Sgl

3. A-h, Sgl, Hh

4. Sgl, A-H, Hh

5. A-H, Hh, Sgl

6. Hh, Sgl, A-H

The curves, A, B & C are respectively:

0

P14- 59

PRS Answer: Helmholtz

Note that the Helmholtz mode creates a very uniform field near the center while the field from the Anti-Helmholtz is zero at the center. The single coil peaks at the coil’s center.

Answer: 6. Helmholtz, Single, Anti-Helmholtz

-2 -1 0 1 2

0Anti-Helmholtz

Single Coil

Helmholtz

Top CoilBottom Coil

Ma

gn

etic

Fie

ld A

mp

litu

de

Distance along the central axis (z/R)

P14- 60

Last Time:Creating Magnetic Fields:

Biot-Savart

P14- 61

The Biot-Savart LawCurrent element of length ds carrying current I produces a magnetic field:

20 ˆ

4 r

dI rsBd

2

ˆx

4o q

r

v r

B

Moving charges are currents too…

P14- 62

Today:3rd Maxwell Equation:

Ampere’s Law

Analog (in use) to Gauss’s Law

P14- 63

Gauss’s Law – The Idea

The total “flux” of field lines penetrating any of these surfaces is the same and depends only on the amount of charge inside

P14- 64

Ampere’s Law: The Idea

In order to have a B field around a loop, there must be current punching through the loop

P14- 65

Ampere’s Law: The Equation

The line integral is around any closed contour bounding an open surface S.

Ienc is current through S:

encId 0sB

enc

S

I d J A

P14- 66

PRS Questions:Ampere’s Law

P14- 67

PRS: Ampere’s Law

Integrating B around the loop shown gives us:

0%

0%

0% 1. a positive number

2. a negative number

3. zero :15

P14- 68

PRS Answer: Ampere’s Law

Answer: 3. Total penetrating current is zero, so

0 0encd I B s

P14- 69

PRS: Ampere’s Law

Integrating B around the loop shown gives us:

0%

0%

0% 1. a positive number

2. a negative number

3. zero 15

P14- 70

PRS Answer: Ampere’s Law

Answer: 2.

Net penetrating current is out of the page, so field is counter-clockwise (opposite path)

0d B s

P14- 71

Biot-Savart vs. Ampere

Biot-Savart Law

general current sourceex: finite wire

wire loop

Ampere’s law

symmetriccurrent source

ex: infinite wireinfinite current sheet

02

ˆ

4

I d

r

s rB

encId 0sB

P14- 72

Applying Ampere’s Law1. Identify regions in which to calculate B field Get

B direction by right hand rule

2. Choose Amperian Loops S: Symmetry B is 0 or constant on the loop!

3. Calculate

4. Calculate current enclosed by loop S

5. Apply Ampere’s Law to solve for B sB

d

encId 0sB

P14- 73

Always True, Occasionally Useful

Like Gauss’s Law,

Ampere’s Law is always true

However, it is only useful for calculation in certain specific situations, involving highly symmetric currents.

Here are examples…

P14- 74

Example: Infinite Wire

I A cylindrical conductor has radius R and a uniform current density with total current I

Find B everywhere

Two regions:

(1) outside wire (r ≥ R)

(2) inside wire (r < R)

P14- 75

Ampere’s Law Example:Infinite Wire

I

I

B

Amperian Loop:

B is Constant & Parallel

I Penetrates

P14- 76

Example: Infinite WireRegion 1: Outside wire (r ≥ R)

dB s

ckwisecounterclo2

0

r

I

B

B ds 2B r

0 encI 0I

Cylindrical symmetry Amperian Circle

B-field counterclockwise

P14- 77

Example: Infinite WireRegion 2: Inside wire (r < R)

2

0 2

rI

R

ckwisecounterclo2 2

0

R

Ir

B

Could also say: 222

; rR

IJAI

R

I

A

IJ encenc

dB s

B ds 2B r

0 encI

P14- 78

Example: Infinite Wire

20

2 R

IrBin

r

IBout

2

0

P14- 79

Group Problem: Non-Uniform Cylindrical Wire

I A cylindrical conductor has radius R and a non-uniform current density with total current:

Find B everywhere

0

RJr

J

P14- 80

Applying Ampere’s LawIn Choosing Amperian Loop:• Study & Follow Symmetry• Determine Field Directions First• Think About Where Field is Zero• Loop Must

• Be Parallel to (Constant) Desired Field• Be Perpendicular to Unknown Fields• Or Be Located in Zero Field

P14- 81

Other Geometries

P14- 82

Two Loops

P14- 83

Two Loops Moved Closer Together

P14- 84

Multiple Wire Loops

P14- 85

Multiple Wire Loops –Solenoid

P14- 86

Demonstration:Long Solenoid

P14- 87

Magnetic Field of Solenoid

loosely wound tightly wound

For ideal solenoid, B is uniform inside & zero outside

Horiz.

comp.

cancel

P14- 88

Magnetic Field of Ideal Solenoid

d d d d d 1 2 3 4

B s = B s B s B s B s

Using Ampere’s law: Think!

along sides 2 and 4

0 along side 3

d

B s

B

n: turn densityencI nlI

0d Bl nlI B s

0

0

nlIB nI

l

/ : # turns/unit lengthn N L

0 0 0Bl

P14- 89

Group Problem: Current Sheet

A sheet of current (infinite in the y & z directions, of thickness 2d in the x direction) carries a uniform current density:

Find B for x > 0

ˆs JJ k

y

P14- 90

Ampere’s Law:Infinite Current Sheet

I

Amperian Loops:

B is Constant & Parallel OR Perpendicular OR Zero

I Penetrates

B

B

P14- 91

Solenoid is Two Current SheetsField outside current sheet should be half of solenoid, with the substitution:

2nI dJ

This is current per unit length (equivalent of , but we don’t have a symbol for it)

P14- 92

=

2 Current Sheets

Ampere’s Law: . encId 0sB

IB

B

X XX

X

X

XX

X

XXX

X

X

XX

X

XXXXXXXXXXXX

B

Long

Circular

Symmetry(Infinite) Current Sheet

Solenoid

Torus

P14- 93

Brief Review Thus Far…

P14- 94

Maxwell’s Equations (So Far)

0Ampere's Law:

Currents make curling Magnetic Fields

enc

C

d I B s

Magnetic Gauss's Law: 0

No Magnetic Monopoles! (No diverging B Fields)S

d B A

0

Gauss's Law:

Electric charges make diverging Electric Fields

in

S

Qd

E A