CH28 Summary: Sources of Magnetic Field
Magnetic field of a moving charge

/current-carrying conductor:  µ0 Idl × rˆ
dB =
Order/Unit/Right-handrule
4π
r2
Magnetic (electric) field of a long
straight (charged) €
conductor:
Forces between current-carrying
conductors:
€
 µ0 qv × rˆ
B=
4π r 2
µ0 I 2a µ0 I
=
4 π xa 2πr
ε λ
E= 0
2πr
| B |=
µ0 I µ0 II' L
F = I' lB = I' L
=
2πr
2πr
€
µ0 Ia 2
Magnetic field of a current loop: Bx =
2( x 2 + a 2 ) 3
€
Ampere’s Law and applications:
 
∫ B • dl = µ0Iencl
B N max =
€
Infinite conductor/Loop
€
€
µ0 NI
2a
Chapter 29: Electromagnetic Induction
•  Faraday’s Law: (changing) Magnetic fieldemf/Current
•
Applications
•  Lenz’s Law: Induced fieldresist change
•  motional emf: moving conductor  motional emf
•  Induced electric fields: From changing magnetic field
•  Induced magenetic fields: From changing electric field
(Displacement current)
•  Maxwell’s equations:
•
Preparing for electromagnetic wave chapter
•  Superconductor and quantum levitation!
Opening question:
•  Our credit card all look alike. How
is the info stored and transmitted?
•  Why cant’ we just scan the card,
but have to swipe it?
•  Power plants convert other forms of
energy into electrical energy. How
is the conversion done?
•  In the following demonstration,
could you tell me what energy is
converted to what other energies?
Induced current and Faraday’s law
•  Faraday’s law:
ε=−
dφ B
dt
Why the “-” sign?
•  Joseph Henry (U.S) obtained the same results in 1831. Why
isn’t it called Henry’s law?
€
•  In the following demonstration, when will there be a
current?
•  How is the magnetic flux calculated?
How to determine the magnetic flux
•  Magnetic flux through a conducting
entity depends on the field
(direction and magnitude), and the
size and orientation of the entity
Uniform B:
€


dφ B = B • dA = B⊥dA = B(dA)cos φ


φ B = ∫ B • dA = ∫ B(dA)cos φ
φ B = BAcos φ
Example: Emf and the current induced in a loop
•  dB/dt=0.020 T/s;A=120cm2, Ω=5.0w;
•  A)What is the induced emf ε, and
induced current I?
•  B)If the loop is replaced by one made of
an insulator, what happens to ε and I?
•  A)
dφ B d(BA)
dB
=
=A
dt
dt
dt
| ε |= (0.02)(0.012)V = 0.24mV
| ε |=
I=
€
ε 0.24mV
=
= 0.048mA
R
5.0Ω
What is direction of an induced emf; What does the sign mean?
•  We need to define the positive direction first. The signs are
always relative to this chosen direction.
•  The direction of induced emf is the direction of induced current
flow
Uniform B:
φ B = BAcos φ
€
We know current also generates
magnetic field. What is the
direction of
This generated magnetic field?
How is it compared with the
change
Of the magnetic field?
Q29.1
A circular loop of wire is in a
region of spatially uniform
magnetic field. The magnetic
field is directed into the plane of
the figure. If the magnetic field
magnitude is constant,
A. the induced emf is clockwise.
B. the induced emf is counterclockwise.
C. the induced emf is zero.
D. The answer depends on the strength of the field.
Q29.2
A circular loop of wire is in a
region of spatially uniform
magnetic field. The magnetic
field is directed into the plane of
the figure. If the magnetic field
magnitude is decreasing,
A. the induced emf is clockwise.
B. the induced emf is counterclockwise.
C. the induced emf is zero.
D. The answer depends on the strength of the field.
Q29.3
A circular loop of wire is placed
next to a long straight wire. The
current I in the long straight
wire is increasing. What current
does this induce in the circular
loop?
A. a clockwise current
B. a counterclockwise current
C. zero current
D. not enough information given to decide
Q29.4
A flexible loop of wire lies in
a uniform magnetic field of
magnitude B directed into the
plane of the picture. The loop
is pulled as shown, reducing
its area. The induced current
A. flows downward through resistor R and is proportional to B.
B. flows upward through resistor R and is proportional to B.
C. flows downward through resistor R and is proportional to B2.
D. flows upward through resistor R and is proportional to B2.
E. none of the above
Slidewire generator
•  A sliding side wire changing the area of the circuit, therefore the
magnetic flux through this circuit. There should be an induced
emf as a result.
•  A)What is the relation between the induced emf and the speed of
the sliding?
•  Define area vector pointing into the page:
dφ B
d(BA)
dA
=−
= −B
dt
dt
dt
dA = (vdt)L
(vdt)L
⇒ ε = −B
= −BLv
dt
ε=−
€
•  B)What if the wire slides to
€
the left?
hint: The area is getting smaller
Work and power in a slidewire generator
•  What is the direction of the force by the field on the moving rod?
•  How would one keep it moving?
•  What’s the applied external power?

 
F = IL × B
ε
BLv
B 2 L2v
F = ILB = LB =
LB =
R
R
R
B 2 L2v 2
Papplied = Fv =
R
•  How is this compared with the power dissipated in the circuit
€
(I2R)? What does this mean?
Work and power in a slidewire generator
•  What is the direction of the force by the field on the moving rod?
•  How would one keep it moving?
•  What’s the applied external power?

 
F = IL × B
ε
BLv
B 2 L2v
F = ILB = LB =
LB =
R
R
R
B 2 L2v 2 I 2
Papplied = Fv =
=
R
R
•  How is this compared with the power dissipated in the circuit
€
(I2R)? What does this mean?
Energy conservation!
A simple alternator: A device that generates and emf
Uniform B field:
φ B = BAcos φ = BA(cosωt)
dφ
ε = − B = BA(ω sin ωt) = ωBAsin ωt
dt
€
Does the direction change?
What if you have many loops of wires?
Move the coil/loop, or the magnet?
DC generator:
What if we want a emf from a rotating loop that always has the
same sign?
Commutator: Flips the connections to outside circuit
exactly when emf flips directions.
Q29.5
The rectangular loop of wire is being
moved to the right at constant
velocity. A constant current I flows
in the long straight wire in the
direction shown. The current
induced in the loop is
A. clockwise and proportional to I.
B. counterclockwise and proportional to I.
C. clockwise and proportional to I2.
D. counterclockwise and proportional to I2.
E. zero.
Lenz’s Law
•  Faraday’s law: Changing Magnetic fieldInduced
emfCurrentInduced magentic field
•  Len’s Law: This induced magnetic field OPPOSES the change!
(Just like humans who don’t like changes. )
Q29.6
The loop of wire is being moved to
the right at constant velocity. A
constant current I flows in the long
straight wire in the direction shown.
The current induced in the loop is
A. clockwise and proportional to I.
B. counterclockwise and proportional to I.
C. clockwise and proportional to I2.
D. counterclockwise and proportional to I2.
E. zero.
Q29.7
The rectangular loop of wire is being
moved to the right at constant
velocity. A constant current I flows in
the long wire in the direction shown.
What are the directions of the
magnetic forces on the left-hand (L)
and right-hand (R) sides of the loop?
A. L: to the left; R: to the left
B. L: to the left; R: to the right
C. L: to the right; R: to the left
D. L: to the right; R: to the right
Motional emf
•  If an isolated conducting rod moves in a magnetic field, what
happens to the charges?
•  Magnetic force will move the charges. Will the movement
continue forever?
The build up of charges
will stop when
ε = vBL
FE = FB
qE = qvB
V
q ab = qvB ⇒
L
Vab = vBL
Source
of emf

 
ε = ∫ (v × B) • dl
General
form
IF the
€ conductors are Stationary only?
Faraday Disk dynamo: rotating conducting disk as a source of emf
Disk rotating at angular velocity of ω; the disk is connected to a brush
and the rod to complete the circuit. What is the current I?
ε
R
ε =?
I=
R0
ε=
ε=

 
(
v
×
B
)
•
d
l =
∫
∫
R0
0
€
0
v(r)Bdr
1
ωrBdr = ωBR0 2 ⇒
2
1
I = ωBR0 2 /R
2
R
∫
R0
Induced electric fields I
•  Magnetic field is 0 outside the solenoid. I changesB
changesemf in the wire loop the measurement of I’ will be
non-zero.
•  Where did the current come from? Who is providing the force the
move the charges?
•  Induced (nonelectrostaticd ) electric field, and it is
non-conservative! (potential has no meaning!)


qε = ∫ (qE ) • dl ⇒


dΦ
ε = ∫ ( E ) • dl = − B ⇒
dt


dΦ
∫ ( E ) • dl = − dtB
Compare with electrostatic situation:
€
∫


E • dl = 0
Induced electric fields: Applications
Opening question: How does credit card work? Why do you have to
Swipe it?
Q29.8
The drawing shows the uniform
magnetic field inside a long, straight
solenoid. The field is directed into the
plane of the drawing, and is increasing.
What is the direction of the electric
force on a positive point charge placed
at point a?
A. to the left
B. to the right
C. straight up
D. straight down
E. misleading question — the electric force at this point is zero
Q29.10
The drawing shows the uniform
magnetic field inside a long, straight
solenoid. The field is directed into the
plane of the drawing, and is increasing.
What is the direction of the electric
force on a positive point charge placed
at point c (at the center of the solenoid)?
A. to the left
B. to the right
C. straight up
D. straight down
E. misleading question —
the electric force at this point is zero
Ampere’s law is not complete without displacement current
• We now know a changing magnetic field induces electric field?
• Can a changing electric field induce magnetic field?
• The electric field between the plates of a charging capacitor IS changing!
 
∫ B • dl = µ0Iencl
OK
€
Not OK! iC=0!
What’s wrong with Ampere’s law!
ΦE = EA =
σ
q
A= ⇒
ε0
ε0
q
dΦE
ε 0 dq
=
=
/ε = i /ε
dt
dt
dt 0 C 0
dΦE
iC = iD = ε 0
dt
d
€
If we add this term to the ampere’s
law,
 
then everything is fine: ∫ B • dl = µ0 (iC + iD )
dΦE
iD = ε 0
dt
Maxwell’s equations
•  Changing E(M) fields induce M(E) field:
•  A varying electric field will give rise to a magnetic field.
•  A varying magnetic field also induces a electric field
•  Maxwell equations:
∫
Gauss’s law

 Qencl
E • dA =
ε0


∫ B • dA = 0
Faraday’s law
∫


dΦB
E • dl = 0 −
dt
Contribution from charge
Ampere’s law
No single magnetic charges!
Contribution from the change
of the other field!
 
dΦ
∫ B • dl = µ0 (iC + ε 0 dtE )
Basis of electromagentic waves: light, radio, x rays, microwave, etc.
€
If you can only learn one slide from this chapter

Changing E

Changing B

Another B arises (Ampere’s law)
 
∫ B • dl = µ (i + ε

Another E arises (Faraday’s

 law)
dΦ
E
•
d
l
=
0
−
∫
dt
0
C
dΦE
)
0
dt
B
€
€
€ FB=ILB
€
€
Fext>=ILB to keep
it from slowing down
(I=ε/R=BLv/R)
Changing B flux
Induced EMF (ε)
Come from work done by Fext
Generate Current
Electric energy
*Superconductivity /Meissner effect/quantum levitation
•  Superconductor is COOL!
•  Not only has ρ=0 (No resistance), but also
has amazing magentic property
•  It expels magnetic flux, causing levitation!
•  http://www.youtube.com/watch?
v=Xts42tFYRRg
•  If an object is coated with a very thin layer
of superconductor material, the magentic
field can manage to penetrate through the
material defects, creating magnetic flux
tubs, and a 3D clamps on the object:
Quantum levitation
•  http://www.youtube.com/watch?
v=VyOtIsnG71U