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- 1
MIT Class:
Sources of Magnetic Fields
Creating Fields: Biot-Savart
Experiment: Magnetic Fields
Ampere’s Law
P14- 2
Magnetic Fields
P14- 3
Gravitational – Electric Fields
Mass m
Charge q (±)
Create:
m
g  G 2 rˆ
r
q
E  ke 2 rˆ
r
Feel:
Fg  mg
FE  qE
Also saw…
Dipole p
Creates:
Feels:
τ  pE
P14- 4
Magnetism – Bar Magnet
Like poles repel, opposite poles attract
P14- 5
Demonstration:
Magnetic Field Lines
from Bar Magnet
P14- 6
Demonstration:
Compass (bar magnet) in
Magnetic Field Lines
from Bar Magnet
P14- 7
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- 8
Bar Magnets Are Dipoles!
• Create Dipole Field
• Rotate to orient with Field
Is there magnetic “mass”
or magnetic “charge?”
NO! Magnetic monopoles do not exist in isolation
P14- 9
Magnetic Monopoles?
Magnetic Dipole
Electric Dipole
p
-q
μ
q
When cut:
2 monopoles (charges)
When cut: 2 dipoles
Magnetic monopoles do not exist in isolation
Another Maxwell’s Equation! (2 of 4)
E

d
A


S
qin
0
Gauss’s Law
 B  dA  0
S
Magnetic Gauss’s LawP14- 10
PRS:
B Field inside a Magnet
P14- 11
PRS: Magnetic Field Lines
The picture shows the field lines outside a
permanent magnet The field lines inside the
magnet point:
0%
0%
0%
0%
0%
0%
1.
2.
3.
4.
5.
6.
Up
Down
Left to right
Right to left
The field inside is zero
I don’t know
15
P14- 12
PRS Answer: Magnetic Field Lines
Answer: 1. They point up inside the magnet
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
P14- 13
Magnetic Field of the Earth
Also a
magnetic
dipole!
North magnetic pole located in southern hemisphere
P14- 14
Fields: Grav., Electric, Magnetic
Mass m
Charge q (±)
Create:
m
g  G 2 rˆ
r
q
E  ke 2 rˆ
r
Feel:
Fg  mg
FE  qE
Dipole p
Create:
E
Feel:
τ  pE
No
Magnetic
Monopoles!
Dipole m
B
τ  μB
P14- 15
What is B?
B is the magnetic field
It has units of Tesla (T)
N
1Tesla  1
Am
This class & next: creating B fields
Next two classes: feeling B fields
P14- 16
How Big is a Tesla?
• Earth’s Field
5 x 10-5 T = 0.5 Gauss
• Brain (at scalp)
~1 fT
• Refrigerator Magnet
• Inside MRI
3T
• Good NMR Magnet
18 T
• Biggest in Lab
150 T (pulsed)
• Biggest in Pulsars
P14- 17
How do we create fields?
P14- 18
What creates fields?
Magnets – more about this later
The Earth How’s that work?
P14- 19
Magnetic Field of the Earth
Also a
magnetic
dipole!
North magnetic pole located in southern hemisphere
(for now)
P14- 20
What creates fields?
Magnets – more about this later
The Earth How’s that work?
Moving charges!
P14- 21
Electric Field Of Point Charge
An electric charge produces an electric field:
r̂
r̂
1
q
ˆ
E
r
2
4 o r
: unit vector directed from q to P
P14- 22
Magnetic Field Of Moving Charge
Moving charge with velocity v produces magnetic field:
P
r̂
mo q v x rˆ
B
2
4 r
r̂ :
m0  4 10 T  m/A
7
unit vector directed
from q to P
permeability of free space
P14- 23
Recall:
Cross Product
P14- 24
Notation Demonstration
X X X X
X X X X
X X X X
X X X X
OUT of page
“Arrow Head”
INTO page
“Arrow Tail”
P14- 25
Cross Product: Magnitude
Computing magnitude of cross product A x B:
C  AxB
C  A B sin 
| C |: area of parallelogram
P14- 26
Cross Product: Direction
Right Hand Rule #1:
C  AxB
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- 27
Cross Product: Signs
ˆi  ˆj  kˆ
ˆj  kˆ  ˆi
ˆj  ˆi  kˆ
kˆ  ˆj  ˆi
kˆ  ˆi  ˆj
ˆi  kˆ  ˆj
Cross Product is Cyclic (left column)
Reversing A & B changes sign (right column)
P14- 28
PRS Questions:
Right Hand Rule
P14- 29
PRS: Cross Product
15
What is the direction of A x B given the following two
vectors?
A
0%
0%
0%
0%
0%
0%
0%
1.
2.
3.
4.
5.
6.
7.
B
up
down
left
right
into page
out of page
Cross product is zero (so no direction)
P14- 30
PRS Answer: Cross Product
Answer: 5. A x B points into the page
A
B
Using your right hand, thumb along A,
fingers along B, palm into page
P14- 31
PRS: Cross Product
What is the direction of A x B given the following two
vectors?
0%
0%
0%
0%
0%
0%
0%
1.
2.
3.
4.
5.
6.
7.
B
A
up
down
left
right
15
into page
out of page
Cross product is zero (so no direction)
P14- 32
PRS Answer: Cross Product
Answer: 6. A x B points out of the page
A
B
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
P14- 33
Continuous charge distributions:
^
Currents & Biot-Savart
P14- 34
From Charges to Currents?
dB  dq v
m
  charge 
s
charge

m
s
v
dq
dB  Ids
P14- 35
The Biot-Savart Law
Current element of length ds carrying current I
produces a magnetic field:

 m 0 I d s  rˆ
dB 
2
4 r
(Shockwave)
P14- 36
The Right-Hand Rule #2
zˆ  ρˆ  φˆ
P14- 37
Animation:
Field Generated by a Moving Charge
P14- 38
Demonstration:
Field Generated by Wire
P14- 39
Example : Coil of Radius R
Consider a coil with radius R and current I
I
I
P
I
Find the magnetic field B at the center (P)
P14- 40
Example : Coil of Radius R
Consider a coil with radius R and current I
I
I
P
I
1) Think about it:
• Legs contribute nothing
I parallel to r
• Ring makes field into page
2) Choose a ds
3) Pick your coordinates
4) Write Biot-Savart
P14- 41
Example : Coil of Radius R
In the circular part of the coil…
d s  ˆr  | d s  ˆr |  ds
I
I
r̂

ds
I
Biot-Savart:
m0 I
dB 
4
m0 I

4
m0 I

4
d s  rˆ
r2
R d
2
R
d
R
m 0 I ds

2
4 r
P14- 42
Example : Coil of Radius R
Consider a coil with radius R and current I
I
I


ds
I
m 0 I d
dB 
4 R
B   dB 
2

0
m0 I

4 R
B
m0 I
2R
m0 I d
4 R
2
m0 I
0 d  4 R  2 
into page
P14- 43
Example : Coil of Radius R
I
I
P
I
 m0 I
B
into page
2R
Notes:
•This is an EASY Biot-Savart problem:
• No vectors involved
•This is what I would expect on exam
P14- 44
PRS Questions:
B fields Generated by Currents
P14- 45
PRS: Biot-Savart
The magnetic field at P points towards the
0%
0%
0%
0%
0%
0%
0%
1.
2.
3.
4.
5.
6.
7.
15
+x direction
+y direction
+z direction
-x direction
-y direction
-z direction
Field is zero (so no direction)
P14- 46
PRS Answer: Biot-Savart
Answer: 3. B(P) is in the +z
direction (out of page)
ĵ
k̂
î
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.
P14- 47
PRS: Bent Wire
15
The magnetic field at P is equal to the field of:
0%
0%
0%
0%
1.
2.
3.
4.
a semicircle
a semicircle plus the field of a long straight wire
a semicircle minus the field of a long straight wire
none of the above
P14- 48
PRS Answer: Bent Wire
Answer: 2. Semicircle + infinite 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
P14- 49
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- 50
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- 51
Experiment:
Magnetic Fields:
Bar Magnets &
Wire Coils
P14- 52
PRS Question:
Part I: B Field from Bar Magnet
P14- 53
PRS: Bar Magnet B Field
Thinking 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.
The red end of the compass points:
0%
0%
0%
0%
0%
0%
0%
0%
1.
2.
3.
4.
5.
6.
7.
8.
Up
Down
N
Right
Left
Up & right
Up & left
Down & right
Down & left
N
S
?
0
P14- 54
PRS Answer: Bar Magnet B Field
Answer: 7. Down & right
N
N
S
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.
P14- 55
Visualization:
Bar Magnet &
Earth’s Magnetic Field
P14- 56
PRS Question:
Part 3: B Field from Helmholtz
P14- 57
PRS: Helmholtz
Identify the three field profiles that you measured as Single (Sgl), 0
Helmholtz (Hh) or Anti-Helmholtz (A-H):
Magnetic Field Amplitude
Top Coil
Bottom Coil
A
B
Helmholtz
Single Coil
0
C
Anti-Helmholtz
-2
-1
0
1
2
Distance along the central axis (z/R)
The curves, A, B & C are respectively:
0%
0%
0%
0%
0%
0%
1.
2.
3.
4.
5.
6.
Sgl, Hh, A-H
Hh, A-H, Sgl
A-h, Sgl, Hh
Sgl, A-H, Hh
A-H, Hh, Sgl
Hh, Sgl, A-H
P14- 58
PRS Answer: Helmholtz
Answer: 6. Helmholtz, Single, Anti-Helmholtz
Magnetic Field Amplitude
Top Coil
Bottom Coil
Helmholtz
Single Coil
0
Anti-Helmholtz
-2
-1
0
1
2
Distance along the central axis (z/R)
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.
P14- 59
Last Time:
Creating Magnetic Fields:
Biot-Savart
P14- 60
The Biot-Savart Law
Current element of length ds carrying current I
produces a magnetic field:

 m 0 I d s  rˆ
dB 
2
4 r
Moving charges are currents too…
mo q v x rˆ
B
2
4 r
P14- 61
Today:
3rd Maxwell Equation:
Ampere’s Law
Analog (in use) to Gauss’s Law
P14- 62
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- 63
Ampere’s Law: The Idea
In order to have
a B field around
a loop, there
must be current
punching
through the loop
P14- 64
Ampere’s Law: The Equation
 
B

d
s

m
I
0 enc

The line integral is
around any closed
contour bounding an
open surface S.
Ienc is current through S:
I enc   J  d A
S
P14- 65
PRS Questions:
Ampere’s Law
P14- 66
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
:15P14- 67
PRS Answer: Ampere’s Law
Answer: 3. Total penetrating current is zero,
so
B

d
s

m
I

0
0 enc

P14- 68
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- 69
PRS Answer: Ampere’s Law
Answer: 2.
B

d
s

0

Net penetrating current is out of the page, so
field is counter-clockwise (opposite path)
P14- 70
Biot-Savart vs. Ampere
BiotSavart
Law
Ampere’s
law
m 0 I d s  rˆ
B
2

4
r
 
 B  d s  m 0 I enc
general
current source
ex: finite wire
wire loop
symmetric
current source
ex: infinite wire
infinite current sheet
P14- 71
Applying Ampere’s Law
1. 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 B  d s
4. Calculate current enclosed by loop S
5. Apply Ampere’s Law to solve for B

 
B

d
s

m
I
0 enc

P14- 72
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- 73
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- 74
Ampere’s Law Example:
Infinite Wire
I
B
I
Amperian Loop:
B is Constant & Parallel
I Penetrates
P14- 75
Example: Infinite Wire
Region 1: Outside wire (r ≥ R)
Cylindrical symmetry 
Amperian Circle
B-field counterclockwise
B

d
s
 B  ds  B  2 r 

 m0 I enc  m0 I
 m0 I
B
counterclo ckwise
2r
P14- 76
Example: Infinite Wire
Region 2: Inside wire (r < R)
 B  d s  B  ds  B  2 r 
 m0 I enc
  r2 
 m0 I 
2 
R 
 m 0 Ir
B
counterclo ckwise
2
2R
 
I
I
I
2
Could also say: J  
;
I

JA


r
enc
enc
2
A R
R 2
P14- 77
Example: Infinite Wire
m 0 Ir
Bin 
2
2R
Bout
m0 I

2r
P14- 78
Group Problem: Non-Uniform
Cylindrical Wire
I
A cylindrical conductor
has radius R and a nonuniform current density
with total current:
R
J  J0
r
Find B everywhere
P14- 79
Applying Ampere’s Law
In 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- 80
Other Geometries
P14- 81
Two Loops
P14- 82
Two Loops Moved Closer
Together
P14- 83
Multiple Wire Loops
P14- 84
Multiple Wire Loops –
Solenoid
P14- 85
Demonstration:
Long Solenoid
P14- 86
Magnetic Field of Solenoid
Horiz.
comp.
cancel
loosely wound
tightly wound
For ideal solenoid, B is uniform inside & zero outside
P14- 87
Magnetic Field of Ideal Solenoid
Using Ampere’s law: Think!

B  d s along sides 2 and 4


B  0 along side 3
 Bd s = Bd s  Bd s  Bd s  Bd s

1
Bl
I enc  nlI

2
3
0

4
0
 0
n: turn density
 B  d s  Bl  m nlI
0
n  N / L : # turns/unit length
B
m 0 nlI
l
 m 0 nI
P14- 88
Group Problem: Current Sheet
y
A sheet of current (infinite
in the y & z directions, of
thickness 2d in the x
direction) carries a
uniform current density:
J s  Jkˆ
Find B for x > 0
P14- 89
Ampere’s Law:
Infinite Current Sheet
B
I
B
Amperian Loops:
B is Constant & Parallel OR Perpendicular OR Zero
I Penetrates
P14- 90
Solenoid is Two Current Sheets
Field outside current sheet
should be half of solenoid,
with the substitution:
nI  2dJ
This is current per unit length
(equivalent of l, but we don’t
have a symbol for it)
P14- 91
 
Ampere’s Law:  B  d s  m 0 I enc
.
B
Long
Circular
Symmetry
I
B
(Infinite) Current Sheet
X
X
X
X
X
X
X
X
X
X
X
X
X
X
B
X
X
Solenoid
=
2 Current
Sheets
X
X
X
X
X
X
X
X
X
X
X
X
Torus
P14- 92
Brief Review Thus Far…
P14- 93
Maxwell’s Equations (So Far)
Gauss's Law:
Qin
 E  dA  
S
0
Electric charges make diverging Electric Fields
Magnetic Gauss's Law:
 B  dA  0
S
No Magnetic Monopoles! (No diverging B Fields)
Ampere's Law:
 Bd s  m I
0 enc
C
Currents make curling Magnetic Fields
P14- 94