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Ben Gurion University of the Negev
www.bgu.ac.il/atomchip, www.bgu.ac.il/nanocenter
Physics 2 for Electrical Engineering
Lecturers: Daniel Rohrlich , Ron Folman
Teaching Assistants: Ben Yellin, Yoav Etzioni
Grader: Gady Afek
Week 8. Faraday’s law – Magnetism in matter • Faraday’s law of
induction • Lenz’s law • “motional emf” • towards Maxwell’s
equations
Sources: Halliday, Resnick and Krane, 5th Edition, Chaps. 34-35.
Magnetism in matter
We classify materials as diamagnetic, paramagnetic or
ferromagnetic according to their magnetic properties.
These three classes of materials differ in their behavior in a
magnetic field.
Magnetism in matter
We classify materials as diamagnetic, paramagnetic or
ferromagnetic according to their magnetic properties.
These three classes of materials differ in their behavior in a
magnetic field.
When a diamagnetic material is placed in a magnetic field, it
generates a weak field in the opposite direction. In this sense,
its behavior is analogous to a dielectric material.
When a paramagnetic material is placed in a magnetic field, it
generates a weak field in the same direction.
Ferromagnetic materials, unlike diamagnets and paramagnets,
may stay magnetized when taken out of a magnetic field.
Magnetism in matter
For diamagnetic and paramagnetic materials, the induced field
is proportional to the applied field; the proportionality constant
is called the magnetic susceptibility and denoted by the letter χ.
In an applied field B, the material generates a field χB and the
total field inside the material is B + χB.
χ for some diamagnetic and paramagnetic materials at 300 K
Paramagnetic
χ
Diamagnetic
χ
Magnetism in matter
A current in an empty solenoid creates a magnetic field B.
What happens to B if we fill the solenoid with (a) aluminum or
(b) copper?
χ for some diamagnetic and paramagnetic materials at 300 K
Paramagnetic
χ
Diamagnetic
χ
Magnetism in matter
It is conventional to define two more vector fields, the
magnetic field H and the magnetization M, as follows:
• μ0H is the magnetic field in vacuum due to currents.
• μ0M is the magnetic field generated by a diamagnetic or
paramagnetic material.
We have defined the magnetic susceptibility by μ0M = χB =
χ(μ0H). Therefore the effective B inside a diamagnetic or
paramagnetic material is
B + χB = μ0H + χ(μ0H) = μ0(1 + χ)H = μH ,
where μ = μ0(1 + χ) is the magnetic permeability of a material.
Magnetism in matter
Rank the following:
• μ0
• μ for a paramagnet
• μ for a diamagnet
Magnetism in matter
A qualitative explanation of diamagnetism: In some atoms,
electrons are paired so that the total magnetic moment of the
atom is zero. The electrons orbit the nucleus in opposite
directions, hence their magnetic dipole moments cancel.
But as we saw in Lecture 6 on the Lorentz force, Slides 47, 48
and 49, an external magnetic field B decreases the cyclotron
frequency ω of one electron and increases ω of the other.
B
B
–
–
ω increases
ω decreases
Magnetism in matter
A qualitative explanation of diamagnetism: In some atoms,
electrons are paired so that the total magnetic moment of the
atom is zero. The electrons orbit the nucleus in opposite
directions, hence their magnetic dipole moments cancel.
The result is a slight magnetic dipole moment opposite to B!
B
B
–
–
ω increases
ω decreases
Magnetism in matter
Living beings are diamagnets because water is diamagnetic.
Frog levitation
Magnetism in matter
A qualitative explanation of paramagnetism: Some atoms have
permanent magnetic moments that interact only weakly with
each other and their orientation is random. But in an external
magnetic field, the atomic moments tend to line up with the
field, though thermal motion may still scatter their orientations.
Some ferromagnetic materials become paramagnetic above a
critical temperature called the Curie temperature, named after
Pierre Curie.
Magnetism in matter
A qualitative explanation of paramagnetism: Some atoms have
permanent magnetic moments that interact only weakly with
each other and their orientation is random. But in an external
magnetic field, the atomic moments tend to line up with the
field, though thermal motion may still scatter their orientations.
This explanation sounds a lot like what happens to a dielectric
material in a capacitor. Why does a dielectric decrease E while
a paramagnetic increases B?
E=0
E≠0
Magnetism in matter
A qualitative explanation of paramagnetism: Some atoms have
permanent magnetic moments that interact only weakly with
each other and their orientation is random. But in an external
magnetic field, the atomic moments tend to line up with the
field, though thermal motion may still scatter their orientations.
Answer: A magnetic dipole μ tends to line up with B just as an
electric dipole p tends to line up with E. But p decreases E
and μ increases B.
E=0
E≠0
Torque on a current loop REVIEW
Whenever 2 moves to the right of 4 , the torque switches
direction. The area vector A always tends to line up with B.
Defining the magnetic
dipole moment of the
current loop to be μ = IA,
we can write τ = μ × B.
IBa
A
2
This is a slide from Lecture
6 on the Lorentz force. Note,
if we line up these current
loops, they will behave like
solenoids with their magnetic
fields parallel to B.
θ
3
θ
B
4
IBa
Magnetism in matter
Since μ increases B, a diamagnetic material (which has μ
antiparallel to B) tends to decrease B while a paramagnetic
material (which has μ parallel to B) tends to increase B, as
stated.
Magnetism in matter
The liquid oxygen in this photograph is suspended between the
poles of a magnet.
Is liquid oxygen
a diamagnetic or
a paramagnetic
material?
Torque on a current loop REVIEW
Whenever 2 moves to the right of 4 , the torque switches
direction. The area vector A always tends to line up with B.
If we integrate τ dθ' starting
from θ' = 0, we get the work
due to the magnetic torque:

WB    d '

  IBA sin  ' d '
IBa
A
2
θ
3
 IBA cos  μ  B ,
so the potential energy U of a magnetic
dipole μ in a field B is U = –μ · B.
This slide from Lecture 6 shows that U = –μ · B.
θ
B
4
IBa
Magnetism in matter
The liquid oxygen in this photograph is suspended between the
poles of a magnet.
Answer: If liquid oxygen were
So is liquid oxygen
diamagnetic, μ · B would be
a diamagnetic or
negative; then U = –μ · B would
a paramagnetic
be minimized by small B and
material?
the liquid would go to where the
B field is weakest. No, liquid
oxygen is paramagnetic! μ · B
is positive, hence U = –μ · B is
minimized by large B and the
liquid goes to where B is
strongest, i.e. between the poles.
Magnetism in matter
The response of a ferromagnetic material to B is not linear.
Therefore the magnetic susceptibility χ is not defined for a
ferromagnetic material such as iron, although B = μH is still
in use.
Every ferromagnetic material divides into magnetized domains
of volume 10–12 – 10–8 m3, each containing 1017 – 1021 atoms.
Magnetism in matter
Every ferromagnetic material divides into magnetized domains
of volume 10–12 – 10–8 m3, each containing 1017 – 1021 atoms.
Experiments show that, in an external magnetic field, domains
parallel to the external field grow larger at the expense of the
other domains. The material may remain magnetized when the
external field is turned off. This effect is called hysteresis.
B
Magnetism in matter
Hysteresis curve of ferromagnetism: Bapp is the applied field,
B is the resultant field.
B
Start
Increasing Bapp
Bapp
Decreasing Bapp
Magnetism in matter
This “memory” of ferromagnetic materials is the basis for
magnetic memory in audio and video tapes, and magnetic
computer disks!
B
Start
Increasing Bapp
Bapp
Decreasing Bapp
Faraday’s law of induction
In 1831, Michael Faraday (in England) and Joseph Henry (in
the U.S.) independently discovered that a changing magnetic
flux ΦB through a conducting circuit induces a current!
Source: UCSC
Faraday’s law of induction
In 1831, Michael Faraday (in England) and Joseph Henry (in
the U.S.) independently discovered that a changing magnetic
flux ΦB through a conducting circuit induces a current!
The sign of the current depends on the sign of dΦB/dt.
Faraday’s law of induction
In 1831, Michael Faraday (in England) and Joseph Henry (in
the U.S.) independently discovered that a changing magnetic
flux ΦB through a conducting circuit induces a current!
The sign of the current depends on the sign of dΦB/dt.
Faraday’s law of induction
In 1831, Michael Faraday (in England) and Joseph Henry (in
the U.S.) independently discovered that a changing magnetic
flux ΦB through a conducting circuit induces a current!
The sign of the current depends on the sign of dΦB/dt.
Faraday’s law of induction
In fact, the induced “emf” E is directly proportional to dΦB/dt.
What is an “emf”? It is short for “electromotive force”, which
is not the correct term because an “emf” is not a force. It has
units of volts.
An “emf” is like a potential, but here, evidently, the concept of
a potential doesn’t work.
Physically, an “emf” is an electric field that is created in a
conductor. A better version of Faraday’s law is

d B
E(r )  dr  
dt
.
Faraday’s law of induction
Example 1: A conducting circuit wound 200 times has a total
resistance of 2.0 Ω. Each winding is a square of side 18 cm. A
uniform magnetic field B is directed perpendicular to the plane
of the circuit. If the field changes linearly from 0.00 to 0.50 T
in 0.80 s, what is the magnitude of (a) the induced “emf” E in
the circuit (b) the induced electric field E, and (c) the induced
current I?
Faraday’s law of induction
Example 1: A conducting circuit wound 200 times has a total
resistance of 2.0 Ω. Each winding is a square of side 18 cm. A
uniform magnetic field B is directed perpendicular to the plane
of the circuit. If the field changes linearly from 0.00 to 0.50 T
in 0.80 s, what is the magnitude of (a) the induced “emf” E in
the circuit (b) the induced electric field E, and (c) the induced
current I?
Answer: (a) We calculate
dΦB/dt = (dB/dt) (200) (area)= (0.625T/s) (200) (18 cm)2
= 4.05 T · m2/s = 4.05 W/s = 4.05 V = E.
(The weber W = T · m2 is the MKS/SI unit of magnetic flux,
and since T = N /(m/s) · C = V · s/m2, we have W/s = V.)
Faraday’s law of induction
Example 1: A conducting circuit wound 200 times has a total
resistance of 2.0 Ω. Each winding is a square of side 18 cm. A
uniform magnetic field B is directed perpendicular to the plane
of the circuit. If the field changes linearly from 0.00 to 0.50 T
in 0.80 s, what is the magnitude of (a) the induced “emf” E in
the circuit (b) the induced electric field E, and (c) the induced
current I?
Answer: (b) The total length of the wire is (200) (4) (0.18 m) =
144 m. From the “emf” = 4.05 V we infer E = 4.05 V/144 m =
0.028 V/m.
(c) The current is I = V/R = 4.05 V/2.0 Ω = 2.0 A.
Faraday’s law of induction
Example 2: Two bulbs are connected to opposite sides of a
loop of wire, as shown. A decreasing magnetic field (confined
to the circular area shown) induces an “emf” in the loop that
causes the two bulbs to light. What happens to the brightness
of each bulb when the switch is closed?
Bulb 1
Bulb 2
B
Faraday’s law of induction
Answer: Bulb 2 stops glowing, since it is shorted out, and
Bulb 1 glows brighter, since it is the only resistance in the
circuit.
Bulb 1
Bulb 2
B
Faraday’s law of induction
Example 3: The conducting bar at the right is pulled right with
force Fapp at speed v. The resistance R is the only resistance in
the circuit. The magnetic field B is constant and perpendicular
to the plane of the circuit. What is the current I and what is the
power applied?
v
L
R
FB
Fapp
I
x
Faraday’s law of induction
Answer: The flux ΦB is BLx, so dΦB/dt = BLv. Thus the
current is I = (BLv)/R. The force FB equals BIL so the power
applied is FBv = BILv =I2R, i.e. the power applied is the power
lost in “Joule heating” of the resistor.
v
L
R
FB
Fapp
I
x
Lenz’s law
Let’s see if we can understand not only the magnitude but also
the sign of the current induced by a changing magnetic field.
The figure below is taken from Example 3 with one change:
The direction of the induced current I is reversed.
v
L
R
FB
Fapp
I
x
Lenz’s law
But if the direction of I is reversed, then so is the direction of
FB; then the bar accelerates to the right, v increases, I increases,
FB increases further without limit, and energy is not conserved.
v
L
FB
Fapp
R
I
x
Lenz’s law
But if the direction of I is reversed, then so is the direction of
FB; then the bar accelerates to the right, v increases, I increases,
FB increases further without limit, and energy is not conserved.
v
L
FB
R
I
x
Lenz’s law
But if the direction of I is reversed, then so is the direction of
FB; then the bar accelerates to the right, v increases, I increases,
FB increases further without limit, and energy is not conserved.
Consider also the direction of the magnetic flux generated by I.
v
L
FB
R
I
x
Lenz’s law
These considerations lead us to conclude, with H. Lenz, that
the current induced in a loop by a changing magnetic flux must
generate an opposite magnetic flux through the loop.
v
L
R
FB
Fapp
I
x
Lenz’s law
Example 1: If we look back at the qualitative explanation of
diamagnetism, we see that electrons in atomic orbits are just
obeying Lenz’s law.
B
B
–
–
ω increases
ω decreases
Lenz’s law
Example 2: The galvanometer indicates a clockwise current
(seen from above). The south pole of the magnet is down. Is
the hand inserting or withdrawing the magnet?
Lenz’s law
Example 2: The galvanometer indicates a clockwise current
(seen from above). The south pole of the magnet is down. Is
the hand inserting or withdrawing the magnet?
Answer: A clockwise current
implies a downward magnetic
flux. So the flux due to the
magnet must be increasing.
The flux from the south pole
of a magnet increases when the
magnet is inserted.
S
N
Lenz’s law
Example 3: A cylindrical magnet of mass M fits neatly into a
very long metal tube with thin steel walls, and slides down it
without friction. The radius of the magnet is r and the strength
of the magnetic field at its top and bottom is B. The magnet
begins falling with acceleration g. (a) Show that the speed of
the magnet approaches a limiting value v. (b) What is the rate
of heat dissipation in the tube, in terms of v and the other data?
Lenz’s law
Answer: (a) The falling magnet induces a circulating current in
the tube. By Lenz’s law, the magnetic field of this current
opposes the falling magnet, until the magnetic force exactly
balances the force of gravity on the magnet, which falls with
constant speed v. (b) Gravity, the only external force on this
system, does work at the rate Mgv. By energy conservation,
this must be the rate of heat dissipation in the tube.
“Motional emf”
A so-called “motional emf” arises when a conductor moves in
a constant magnetic field. Thus the moving bar (below) is an
example of a “motional emf”. But a “motional emf” can arise
also from the Lorentz force without any magnetic induction.
v
L
R
FB
Fapp
I
x
“Motional emf”
Example 1: A conducting strip of length L moves sideways
with constant velocity v through a constant B pointing out of
the screen. What is the potential difference ΔV between the
two ends of the strip?
B
++
L
v
FB
–
–
“Motional emf”
Answer: At equilibrium, the force on charges anywhere in the
strip must vanish, i.e. E = vB as in Hall effect. The potential
difference is then ΔV = EL = vBL.
B
++
L
v
FB
–
–
“Motional emf”
Example 2: A conducting strip of length L rotates around a
point O with constant angular frequency ω, in a constant B
pointing out of the screen. What is the potential difference
ΔV between the two ends of the strip?
v
r
dr
L
O
“Motional emf”
Answer: An electron in an element dr of the conducting strip
is subject to a centripetal magnetic force evB which must be
balanced by an electric force eE = evB = eωrB. (Note v is not
uniform along the strip.) Integrating E(r)dr along the strip, we
obtain V 
L
0
L
0
E (r )dr  B rdr  BL2 /2 .
v
r
dr
L
O
Towards Maxwell’s equations
The set of four fundamental equations for E and B,
 E  dA   0
q

 B  dr   0 I
 B  dA  0 ,
d
E  dr    B
dt
(Gauss’s law)
(Faraday’s law)
(Ampère’s law)
together with the Lorentz force law FEM = q (E + v × B), sum
up everything we have learned so far about electromagnetism!
Towards Maxwell’s equations
The set of four fundamental equations for E and B,
 E  dA   0
q

 B  dr   0 I
 B  dA  0 ,
d
E  dr    B
dt
(Gauss’s law)
(Faraday’s law)
(Ampère’s law)
are similar to the famous equations named after J. C. Maxwell
– “Maxwell’s equations” – describing all of electromagnetism.
But they are not yet Maxwell’s equations!
Towards Maxwell’s equations
The set of four fundamental equations for E and B,
 E  dA   0
q

 B  dr   0 I
 B  dA  0 ,
d
E  dr    B
dt
(Gauss’s law)
(Faraday’s law)
(Ampère’s law)
include one equation with an error that Maxwell discovered
and corrected. What is the error?