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Lesson 5
Representing Fields
Geometrically
Class 13
Today, we will:
• review characteristics of field lines and contours
• learn more about electric field lines and
contours of point charges
• learn more about magnetic fields and contours
of long, straight wires
Section 1
Review of Field Lines from
Lesson 2
Review from Lesson 2
•For point charges:
–Electric fields are obtained from
aligning threads
–Magnetic fields are obtained from
aligning stubs.
Review from Lesson 2
•For point charges:
–Electric fields are
radially outward from a
+ source and inward
toward a – source.
–Magnetic fields form
circles around the path
of the source. The
direction is given by a
right-hand rule.
Field Lines and Threads
• Align some threads in a
box with the ends of the
box perpendicular to the
threads. The height of the
box is ℓ.
A
ℓ
Field Lines and Threads
• Align some threads in a
box with the ends of the
box perpendicular to the
threads. The height of the
box is ℓ.
• Each thread is part of one
field line when aligned.
A
ℓ
Field Lines and Threads
• Align some threads in a
box with the ends of the
box perpendicular to the
threads. The height of the
box is ℓ.
A
ℓ
• Each thread is part of one
field line when aligned.
• The force is proportional to
the density of threads time
the length of the threads.
F  
Field Lines and Threads
• Align some threads in a
box with the ends of the
box perpendicular to the
threads. The height of the
box is ℓ.
A
ℓ
• Each thread is part of one
field line when aligned.
• The force is proportional to
N
N
N
the number of field lines
F     

V
A
A
per unit area.
Field Lines
• To find the strength of a field from the field
lines:
1)Take a small section of a contour surface of
area A.
2)Count the number of field lines
passing through: N.
A
3) The field is proportional to N/A.
Field Contours
• Elements of the field contour are surfaces that
1) are perpendicular to the field lines.
2) are spaced so that the field is stronger
where the surfaces are close together.
Field Contours
• To find the strength of a field from the field
contour:
1)Take a small section of a field line of length L.
2)Count the number of contour surfaces
pierced by the line segment: N.
3) The field is proportional to N/L.
Section 2
The Electric Field of Point
Charges
Electric Field
• The fundamental source of electric field is a
point charge.
• All electric fields arise from a collection of point
charges.
• We will only consider static electric fields for
now.
Electric Field of a Point Charge
• Radially outward for positive charges.
• Radially inward for negative charges.
• Uniform in all directions in space.
• Number of field lines is proportional to the
magnitude of the charge.
Electric Field of a Point Charge
Electric Field of a Point Charge
Electric Field of a Point Charge
“Flat” Representation
Electric Field of a Point Charge
“Flat” Representation
Coulomb’s Law = Electric Field Lines
1
q
1 1
E

2
40 r
40 12
N
N
4
Ek k
2  k
2
A
4


1
4 r
Let’s use SI Units. If q =1C and r =1m
and there are 4 field lines per coulomb:
1
4
1
E
k
k 
40
4
4 0
1 N
E
4 0 A
…but only when we
draw 4 lines per C
Coulomb’s Law = Electric Field Lines
If q = 1 there are 4 field lines.
1 4
1

At r  1, E 
2
4 0 4 1
40
If q = 3 there are 12 field lines.
1 12
3
At r  1, E 

2
4 0 4 1
40
→
Eq
Coulomb’s Law = Electric Field Lines
There are twelve field
lines.
1 12
3

At r  1, E 
2
4 0 4 1
40
1 12
1 3

At r  2, E 
2
4 0 4 2
4 40
→
1
E 2
r
Coulomb’s Law = Electric Field Lines
“Coulomb’s Law is built into
the geometry of the electric
field lines.”
All we had to do is choose
the right number of lines, let
them point radially outward
or inward, and let them be
uniformly distributed.
Electric Field Contours
Electrical field contours are
sets of spheres. We have to
compute the separation
distances between surfaces
and place them in the right
places “by hand.”
“Electric field contours are
unconstrained by geometry.”
Section 3
The Magnetic Field of Long,
Straight Wires
Magnetic Field
• The fundamental source of magnetic field is an
infinitely long, very thin, current-carrying wire.
• Not all magnetic fields arise from such wires,
but their fields are simple enough we’ll use
them anyway.
Magnetic Field of a Long Wire
• We calculated the field of a wire using stubs,
but for now we’ll consider this an experimental
result independent of Coulomb’s Law.
 0i
B
,  0  4  10 7 Tm / A
2r
1
0 
2
 0c
“permeability of free space”
Magnetic Field Lines of a Long Wire
• Magnetic field lines form circles around a wire.
The direction is given by the right-hand rule.
Magnetic Field Lines of a Long Wire
• Put your thumb in the direction of the current
and the magnetic field lines will go around the
wire in the direction of your fingers.
Magnetic Field Lines
Magnetic field lines are sets
of circles. We have to
compute the separation
distances between lines and
place them in the right
places “by hand.”
“Magnetic field lines are
unconstrained by geometry.”
Magnetic Field Contours of a Long Wire
• Magnetic field contours are sets of half-planes
having one edge on the wire.
Magnetic Field in Two Dimensions
•We choose to draw three surfaces per ampere.
•We attach direction arrows to the surfaces in the
direction of the field.
Wire Law = Magnetic Field Contours
0 i 0 1
B

2 r 2 1
N
N
3
B  k  k
 k
L
2 r
2 1
Let’s use SI Units. If i =1A and r =1m
and there are 3 surfaces per ampere:
0
0
3
B
 k
 k 
2
2
3
B
0 N
3 L
…but only when we
draw 3 lines per A
Wire Law = Magnetic Field Contours
If i = 1 there are 3 field surfaces.
0 3
0
At r  1, B 

3 2 1 2
If i = 2 there are 6 field surfaces.
0 6
0
At r  1, B 

3 2 1 
→
Bi
Wire Law = Magnetic Field Contours
There are six field
surfaces.
0 6
0

At r  1, B 
3 2 1 
At
→
0
0
6
r  2, B 

3 2  2 2
1
B
r
Wire Law = Magnetic Field Contours
“The Wire Law is built into the
geometry of the magnetic
field contours.”
All we had to do is choose the
right number of surfaces,
orient them correctly, and let
them be uniformly distributed.
Class 14
Today, we will:
• learn how to draw the total electric field of two
point charges.
• find that the electric field is like a single point in
the near field and the far field.
• use symmetry to find the electric field lines of
charged spheres, cylinders, and palnes.
Section 4
The Electric Field of Multiple
Charges
Electric Fields - Threads
A + charge and a ̶ charge each emit threads.
Electric Fields - Threads
The threads of both charges combine to give a total force
on a field particle.
Electric Field Lines
Electric field lines point in the direction of the net force.
Electric Field Lines
Take one vector pointing in the direction of the electric
field near a positive charge.
Electric Field Lines
Then attach the head of a second vector to the tail of the
first vector.
Electric Field Lines
Continue this process over and over.
Electric Field Lines
This forms an electric field line for the pair of charges.
Electric Field Lines
The direction of the electric field is tangent to the
electric field line as before.
Electric Field Lines
Recall that the electric field is stronger where the lines
are closer together.
stronger field
weaker field
How to Draw Electric Field
Lines of Multiple Charges
Two Point Charges
Let’s take charges of +2 C and -2 C.
Near Field
What will the field be like near the positive
charge?
Near Field
Hint: Don’t draw field lines that go directly
toward or away from another charge.
Near Field
What will the field be like near the negative
charge?
Near Field
Far Field
What will the field be like far from the two
charges?
Far Field
There is no field (or at least it’s very weak).
Two Point Charges
Charges of +2 C and -2 C.
Two Point Charges
Charges of +2 C and -2 C.
Two Point Charges
Charges of +2 C and -2 C.
Two Point Charges
Charges of +2 C and -2 C.
Two Point Charges
Charges of +2 C and -2 C.
Two Point Charges
Charges of +2 C and -2 C.
Two Point Charges
Let’s go to ActivPhysics – so we can adjust
parameters easily.
Remember to consider the near field, the far
field, and then the intermediate field.
http://wps.aw.com/aw_knight_physics_1
Two Point Charges
Two Point Charges
Two Point Charges
Two Point Charges
Two Point Charges
Two Point Charges
Clicker Question 1
Is this picture OK?
A – yes B -- no
+1
–1
Clicker Question 1
Is this picture OK?
A – yes B -- no
+1
–1
•Field lines can not cross.
•The near field is incorrect.
Clicker Question 2
Is this picture OK?
A – yes B -- no
+2
–1
Clicker Question 2
Is this picture OK?
A – yes B -- no
+2
–1
•The far field is incorrect.
•The near field isn’t drawn well.
Section 5
Electric Field Lines and
Symmetry
Symmetry
What can we say about the electric field lines of a
sphere?
Spherically Symmetric Charge
How do the electric field lines of a spherically
symmetric charge compare to a point charge?
If the charge is the same, the total number of field
lines is the same.
Spherically Symmetric Charge
Outside of the sphere:
N
1 q
Ek 
A 40 r 2
Cylindrically Symmetric Charge
What do the field lines of a cylindrically symmetric
charge look like?
Cylindrically Symmetric Charge
What do the field lines of a cylindrically symmetric
charge look like?
Cylindrically Symmetric Charge
How do the electric field lines of a cylindrically
symmetric charge compare to a point charge?
If the charge in length L is the same as the point
charge, the total number of field lines coming from
length L is the same as from the point charge.
Cylindrically Symmetric Charge
point charge with
a charge q.
infinitely long rod with a
charge q on a segment of
length L
q

L
Cylindrically Symmetric Charge
N
N
Ek k
A
4 r 2
1 q
E
40 r 2
kN 
q
0
←same N →
E outside cylinder→
Draw a sphere around the
point charge
N
kN

A 2 r L
q 1
E
 0 L 2 r
Ek
E
1

2  0 r
Cylindrically Symmetric Charge
N
N
Ek k
A
4 r 2
1 q
E
40 r 2
kN 
q
0
←same N →
E outside cylinder→
N
kN

A 2 r L
q 1
E
 0 L 2 r
Ek
E
1
q


L
2  0 r
Draw a cylindrical can of length L
around the cylinder
Planar Symmetric Charge
What do the field lines of an infinite plane of
charge look like?
Planar Symmetric Charge
What do the field lines of an infinite plane of
charge look like?
Planar Symmetric Charge
How do the electric field lines of a infinite plane of
charge compare to a point charge?
If the charge in area A is the same as the point
charge, the total number of field lines coming from
area A is the same as from the point charge.
Cylindrically Symmetric Charge
point charge with
a charge q.
infinite plane with a charge q
on a segment of area A
q

A
Planar Symmetric Charge
N
N
Ek k
A
4 r 2
1 q
E
40 r 2
kN 
q
←same N →
but N/2 lines
on each side
of the plane!
0
Use a spherical surface again
N /2
Ek
A
q
E
2 0 A
E

2 0
Planar Symmetric Charge
N
N
Ek k
A
4 r 2
1 q
E
40 r 2
kN 
q
0
←same N →
but N/2 lines
on each side
of the plane!
N /2
Ek
A
q
E
2 0 A
E

2 0
Threads (point) vs. Field Lines (bulk)
very small, move radially
outward from charges
move at the speed of light
go in straight lines
can not be destroyed, but
their effects can be
cancelled by other threads
start on positive charges and end on
negative charges – or go off to
infinity
changes “move” at the speed of
light
bend around, following the
direction of the force
do not exist in regions where there
is no net force
magnetic fields result from magnetic fields result from
motion correction to
1)currents (unrelated to electric
threads
fields) or from 2)changing electric
fields – but we won’t see the
second case until later.
Class 15
Today, we will:
• learn to visualize magnetic field lines for two
wires.
• learn to visualize electric and magnetic field
contours for two charges or wires.
Symmetry and Field Lines
1) If we know how much charge an object has, we
know how many field lines there are.
2) If there is enough symmetry (spheres, cylinders,
planes), we know what the field lines look like.
3) If we know where all the field lines are, we can
measure the number of field lines per unit area
anywhere in space, so we know the electric field
everywhere!
Section 6
Magnetic Field Lines and Electric
Field Contours of Multiple Sources
What about Magnetic Field Contours?
Their pictures are not exactly the same as electric field
lines, but if we want close instead of exact, we can draw
them the same way.
What about Magnetic Field Contours?
Same charges (left) and same currents (right).
electric field lines
electric field contours
magnetic field contours
magnetic field lines
What about Magnetic Field Contours?
Opposite charges (left) and opposite currents (right).
electric field lines
electric field contours
magnetic field contours
magnetic field lines
Magnetic Field Contours
Conclusion: If you can draw electric field lines for two
charges, you can draw magnetic field contours for two
currents!
electric field lines
electric field contours
magnetic field contours
magnetic field lines
What about Electric Field Contours
and Magnetic Field Lines?
electric field lines
electric field contours
magnetic field contours
magnetic field lines
How do we make electric field
contours?
Start with the electric field lines.
How do we make electric field
contours?
Add spherical contours in the near field.
How do we make electric field
contours?
The next surfaces are deformed – closer where the field
is stronger.
How do we make electric field
contours?
Then a single surface passes around both charges.
How do we make electric field
contours?
The surfaces become more spherical…
How do we make electric field
contours?
until they become spheres in the far field.
Now take the case of opposite
charges.
Start with the electric field lines.
Now take the case of opposite
charges.
Add spherical contours in the near field.
Now take the case of opposite
charges.
The next surfaces are deformed – closer where the field
is stronger.
Now take the case of opposite
charges.
The next surfaces are deformed – closer where the field
is stronger.
Now take the case of opposite
charges.
There are no surfaces nor lines at infinity in this case.
Diagrams for magnetic field look similar to
those of the electric field.
The field lines and contours reverse roles, however!
Diagrams for magnetic field look similar to
those of the electric field.
Two equal currents going in the same direction.
field contour
field lines
Diagrams for magnetic field look similar to
those of the electric field.
Two equal currents going in opposite directions.
field contour
field lines