Magnetic fields and magnetic
forces
• Han Christian Oersted/André Ampere – discovered the
relationship between moving charges and magnetism
• Michael Faraday/Joseph Henry – discovered that moving a
magnet near a conducting loop can cause a current in the
loop
• Magnetic fields are produced by electric currents.
• The Lorentz force: F=qvxB
• SI unit: Testa (T)
• If the charge is moving in a region where E and B fields are
present:
Magnetic field sources
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The magnetic field
lines around a
long wire
The magnetic
field lines of a
current loop
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Soleniod
Bar magnet
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The bar magnet
The earth
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The bar magnet and the earth
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http://www.timed.jhuapl.edu/WWW/science/objectives.php
Imager of Sprites and Upper
Atmospheric Lightning (ISUAL)
Imager of Sprites and Upper
Atmospheric Lightning (ISUAL)
The Lorentz force
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Magnetic force on moving charge
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Mass spectrometer
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e/m experiment
http://physics.csustan.edu/GENERAL/Ian/GeneralPhysicsIIlabs/EoverM/movies/Path.htm
Magnetic force on a currentcarrying wire
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What is generated in the wire?
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Magnetic flux
• Magnetic flux is the product of the average
magnetic field and the perpendicular area that it
penetrates.
   B  dA
d
B
dA
Magnetic flux density
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Gauss’ law of magnetism
The net magnetic flux out of any closed
surface is zero.
What is the physical significance of this
statement?
Units:
1. Magnetic field B: 1T=1Ns/Cm
2. Magnetic flux : 1W=1Tm2
Motion of a charged particles in a
magnetic field
• Two positive ions having the same charge q, but different masses
m1 and m2, are accelerated from rest through a potential difference
V. They then enter a region where there is a uniform magnetic field
B normal to the plane of the trajectory. Show that if the beam
entered the magnetic field along the x-axis, the value of the ycoordinate for each ion at any time t is approximately
 q 
y  Bx 

 8mV 
2
1
2
provided y is remains much smaller than x.
• Can this be used for isotope separation?
Magnetic force on a currentcarrying conductor
The force on all of the moving charges:
F  nAl qvd B   nqvd AlB 
but
J  nqvd
 F  IlB
In general
F  Il  B
dF  Idl  B
differential form
A
l
vd
An electromagnetic rail gun
• A conducting bar with mass m and length L slides over a horizontal
rails that are connected to a voltage source. The voltage source
maintains a constant current I in the rails and bar; and a constant,
uniform, vertical magnetic field B fills the region between the rails.
– Find the magnitude and direction of the net force on the conducting bar.
Ignore friction, air resistance and electrical resistance.
– If the bar has a mass m, find the distance d that the bar must move
along the rails from rest to attain speed v.
– It has been suggested that rail guns based on this principle could
accelerate payloads into earth orbit or beyond. Find the distance the bar
must travel along the rails if it is to reach the escape speed for the earth.
Let B=0.5T, I=2000A, m=25kg and L=0.5m.
I
B
Force and torque on a current loop
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Hall effect
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The magnetic dipole moment
(magnetic moment)
  IA
Torque
τ  μB
Where is the direction
of the solenoid’s
tendency of rotation?
I
B
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Sources of magnetic field
• The magnetic field produced by a moving charge is
proportional to the
– charge
– velocity of the charge
– inverse of the square of the distance
0 q v sin 
B
4
r2
0 qv  rˆ
B
4 r 2
 0  4 10 7 TmA1 (permeability constant)
fr
q
v
s
i
o
e
u
rl
d
c
p
e
o
p
i
o
in
t
n
Magnetic field of a current element
dQ  nqAdl
0 n q vd Adl sin 
dB 
4
r2
0 Idl sin 
dB 
4
r2
 0 Idl  rˆ
dB 
4 r 2
0 Idl  rˆ
B
4  r 2
source
point
dl
I
r̂
field point
r
dB
Biot-Savart law
• For an infinitely long straight wire, the
magnetic field at a distance x from the
wire is given by
0 I
B
2x
• Ampere’s law
 B  dl   I
0
Force between parallel conductors
0 I
B
2r
 0 II ' l
F  I ' lB 
2r
F  0 II '

l
2r
r
x
x
I’
I
B
One ampere is that unvarying current that, if
present in each of tow parallel conductors of
infinite length and one meter apart in empty
space, causes each conductor to experience a
force of exactly 2x10-7 Nm-1.
Magnetic field of a circular current
loop
z
dl
r̂
I
dBy 
dBz


dBy
x
 0 Idl  rˆ
dB 
4 r 2
Let a be the radius of the ring.
dB 
0 I
dl
4 x 2  a 2 2
y
0 I
dl
sin 
2
2
4 x  a 
0 I
dl
dBz 
cos 
4 x 2  a 2 
I
1
a
By  0
4 x 2  a 2  2
2
By 
 0 Ia 2
2x  a
2
x
a

1
2
 dl

3
2 2
What is B at the center of the ring?
If there are N rings, what is B at the
center of the rings?
The magnetic field at a distance r from a
conductor has a magnitude
I
B
.
B
dl
.
dl
B
0
2r
0 I
 B  dl   Bdl  B  dl  2r 2r   0 I
 B  dl   B 1dl   B  dl    I
0
  B  dl   0 I
d
4
c
r2
1
.r
3
1
b
c
d
a
b
c
 B  dl   B  dl   B  dl  
a
B  dl   B  dl
d
b 2 a
b
c
a
b
b
c
a
b
 B  dl   0dl  
 B  dl   0dl  
d
a
c
d
d
a
c
d
B3dl   0dl   B1dl
B3dl   0dl   B1dl
0 I
 0 I
 B  dl  2r1 2r1   2r2 2r2   0
Take note:
Even though there is a magnetic field everywhere along the
integration path, the line integral is zero if there is no current
passing through the area bounded by the path.
Ampere’s law
.
 B  dl  0 I enclosed
x
.
x
.
path of
integration
x
x
Curl your fingers of your right hand around the
integration path so that they curl in the
direction of integration. Then your right thumb
indicates the positive current direction.
Applications of Ampere’s law
A long, straight, solid cylinder, oriented with its axis in the z-direction,
carries a current whose current density is J. The current density, although
symmetrical about the cylinder axis, is not constant but varies according to
the relation
2
2I 0
J 2
a
J0
 r
1   
  a 

kˆ

for ra
for ra
where a is the radius of the cylinder, r is the radial distance from the
cylinder axis, and Io is a constant having units of amperes.
a) Show that Iois the total current passing through the entire cross section
the wire.
b) Using Ampere’s law, derive an expression for the magnitude of the
magnetic field B in the region ra.
c) Obtain and expression for the current I contained in the circular cross
section of radius ra and centered at the cylinder axis.
d) Using Ampere’s law, derive an expression for the magnitude of the
magnetic field B in the region ra.
Applications of Ampere’s law
The figure below shows an end view of two long, parallel wires
perpendicular to the xy-plane, each carrying a current I but in
opposite direction. Derive the expression for the magnitude of B at
any point on the x-axis.
.
a
P
x
a
x
Applications of Ampere’s law
A circular loop has radius R and carries current I2 in a clockwise
direction. The center of the loop is a distance D above a long,
straight wire. What are the magnitude and direction of the current I1
in the wire if the magnetic field at the center of the loop is zero?
R
I2
D
I1
Study the examples in the book!
Field inside a long cylindrical
conductor
B
R
r
I
B
Amperian loop
 B  dl   I
0 enclosed
 r2 
B2r   0  I 2 
 R 
B
0 I r
2 R 2
R
r
Field of a
solenoid
Let n be the number of turns.
 B  dl   I
0 enclosed
BL  0 nLI
B  0 nI
Find the magnetic field of a
toroidal solenoid.
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Magnetic materials
But

I
e
L
  IA
e
ev
I 
T 2r
ev
evr
2


r  
2r
2
L  r  mv
L  mvr
e

L
2m
The dipole moment’s
component in a
particular direction is
an integral multiple of

h
2
h  6.626 1034 Js
When we speak of the magnitude of a
magnetic moment, we mean the
“maximum component in a given
direction”.
 aligned with B means that  has its
maximum possible component in the
direction of B.
The Bohr magneton
If
L
h
2

eh
 9.274 10  24 Am 2
4m
Magnetization M 
μ
I
i
i
V
Paramagnetism
The magnetization of the material is proportional to the
applied magnetic field in which the material is placed.
B  B 0  μ 0M

Curie’s law
B
M C
T
Mmagnetization
CCurie’s constant
Ttemperature (K)
Bmagnetic field
For a given ferromagnetic material the long range order abruptly
disappears at a certain temperature called the Curie temperature.
The relative permeability:
  K m 0
For common paramagnetic materials, Km
varies from 1.00001 to 1.003.
The magnetic susceptibility:
m  Km 1
The magnetic susceptibility is the amount
by which the relative permeability differs
from unity.
Material
Curie temperature
(K)
Fe
1043
Co
1388
Ni
627
Gd
293
Dy
85
CrBr3
37
Au2MnAl
200
Cu2MnAl
630
Cu2MnIn
500
EuO
77
EuS
16.5
MnAs
318
MnBi
670
GdCl3
2.2
Fe2B
1015
MnB
578
Data from F. Keffer, Handbuch der Physik, 18, pt. 2, New York: Springer-Verlag,
1966 and P. Heller, Rep. Progr. Phys., 30, (pt II), 731 (1967)
Material
m
Iron ammonium alum
66
Uranium
40
Platinum
26
Aluminum
2.2
Sodium
0.72
Oxygen gas
0.19
Bismuth
-16.6
Mercury
-2.9
Silver
-2.6
Carbon (diamond)
-2.1
Lead
-1.8
Sodium chloride
-1.4
Copper
-1.0
Young &Freedman, University Physics 11th ed.,
p.1089
Diamagnetism
The orbital motion of
electrons creates tiny
atomic current loops,
which produce magnetic
fields. When an external
magnetic field is applied
to a material, these
current loops will tend to
align in such a way as to
oppose the applied field.
Diamagnetism is the
residual magnetic
behavior when materials
are neither paramagnetic
nor ferromagnetic.
Ferromagnetism
Strong interactions between magnetic moments cause to line up
parallel to each other in regions called magnetic domains.
Hysteresis
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Average
magnetic dipole
moment per
atom of iron
A piece of iron has a
magnetization
M=6.5x104 A/m. Find
the average magnetic
dipole moment per
atom in this piece of
iron. Express your
answer in Bohr
magnetons and in Am2.
The density of iron is
7.8x103kg/m3. The
atomic mass of iron is
55.845g/mol.
Electromagnetic induction
• Results from experiments
S
N
electromagnet
galvanometer
– When there is no current in the
electromagnet, so that B=0, the
galvanometer shows no current.
– When the electromagnet is turned
on, there is a momentary current
through the meter as B increases.
– When B levels off at a steady
value, the current drops to zero,
no matter how large B is.
Electromagnetic induction
• Results from experiments
S
N
electromagnet
galvanometer
– With the coil in a horizontal plane,
we squeeze it so as to decrease
the cross sectional area of the
coil. The meter detects current
only during the deformation, not
before or after. When we increase
the area to return the coil to its
original shape, there is current in
the opposite direction, but only
while the area of the coil is
changing.
Electromagnetic induction
• Results from experiments
S
N
electromagnet
galvanometer
– If we rotate the coil a few degrees
about a horizontal axis, the meter
detects a current during the
rotation, in the same direction as
when we decreased the area.
When we rotate the coil back, there
is a current in the opposite direction
during this rotation.
– If we jerk the coil out of the
magnetic field, there is a current
during the motion, in the same
direction as when we decreased
the area.
Electromagnetic induction
• Results from experiments
S
N
electromagnet
galvanometer
– If we decrease the number of
turns in the coil by unwinding
one or more turns, there is a
current during the unwinding,
in the same direction as when
we decreased the area. If we
wind more turns onto the coil,
there is a current in the
opposite direction during the
winding.
Electromagnetic induction
• Result from experiments
S
N
electromagnet
– When the magnet is turned off, there
is a momentary current in the
direction opposite the current when it
was turned on.
– The faster we carry out any of these
changes, the greater is the current.
– If all these experiments are repeated
with a coil that has the same shape
but different material and different
resistance, the current in each case
is inversely proportional to the total
circuit resistance.
galvanometer
What is common in these results?
Electromagnetic induction
Faraday’s law: The induced emf in a closed loop equals
the negative of the time rate of change of magnetic flux through
the loop.
d B
 
dt
 B   B  dA
Any change in the magnetic environment of a coil of wire will
cause a voltage (emf) to be "induced" in the coil. No matter how
the change is produced, the voltage will be generated. The
change could be produced by changing the magnetic field
strength, moving a magnet toward or away from the coil, moving
the coil into or out of the magnetic field, rotating the coil relative
to the magnet, etc.
For N turns,
d B
  N
dt
Direction of induced emf
Increasing
B
A
<0
Decreasing
B
A
>0
• Define the direction of the vector
area A.
• From the directions of A and the
magnetic field B, determine the
sign of the magnetic flux B and its
rate of change.
• Determine the sign of the induced
emf or current. If the flux is
increasing, so dB/dt is positive,
then the induced emf or current is
negative.
• If the flux is decreasing, dB/dt is
negative and the induced emf or
current is positive.
Direction of induced emf
A
>0
A
<0
• Determine the direction of the
induced emf or current using
your right hand. Curl the fingers
of your right hand around the A
Increasing vector, with your right thumb in
B
the direction of A. If the
induced emf or current in the
circuit is positive, it is in the
Decreasing same direction as your curled
B
fingers. If the induced emf or
current is negative, it is in the
opposite direction.
Faraday’s law
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Faraday’s law
Lenz’s law: The direction of any magnetic induction effect
is such as to oppose the cause of the effect.
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Examples
Two coupled circuits, A and B, are situated
as shown below. Use Lenz’s law to
determine the direction of the induced
current in resistor ab when (a) coil B is
R
brought closer to coil A with the switch
closed, (b) the resistance of R is decreased
while the switch remains closed, and (c)
switch S is opened.
An alternator is a device that generates an emf.
A rectangular loop is made to rotate with a
constant angular velocity w about the axis
shown below. The magnetic field B is uniform
and constant. At time t=0, f=0, determine the
induced emf.
A
B
S
a b
Examples
Consider a motor with a square coil 10cm on a side with
500 turns of wire. If the magnitude of the B field is 0.2T, at
what rotation speed will the average back emf of the
motor be 112V? The back emf of a motor is the emf
induced by changing magnetic flux through its rotating
coil.
A conducting disk with radius R lies in the xy-plane and
rotates with constant velocity w about the z-axis. The disk
is in a uniform, constant B field parallel to the z-axis. Find
the induced emf between the center and the rim of the
disk.
Examples
Consider a U-shaped conductor in a uniform Bfield. If a metal with length L is put across the arms xx
of the conductor, forming a circuit, and move the xx
x
rod to the right with constant velocity v, find the
magnitude and direction of the resulting emf.
x x x x x
x x x x x
x x x x x
x x x x x
x x x x x
A cardboard tube is wrapped with two windings of insulated
wire wound in opposite directions. Terminals a and b of
winding A may be connected to a battery through a reversing
switch. Where is the direction of the induced current in the
resistor R if a) the current in winding A is from a to b and is
increasing? b) the current is from b to a and decreasing? c)
a
b Winding A
the current is from b to a and increasing?
Winding B
v
Examples
A long wire carries a constant current I. A metal bar with
length L is moving at constant velocity v. Calculate the emf
induced in the bar? Which point is at higher potential? What
is the magnitude of the induced current if the bar is replaced
by a rectangular wire loop?
I
I
v
v
Motional electromotive force
a
xxxxxxxxxxxxxxxx
xI x x x x x x x x x x x x x xvx
x x x x x x x x x x x xL x x x x
xxxxxxxxxxxxxxxx
b
Motional electromotive force:
  vBL
d  v  B  dL
   v  B  dL
This equation can be used
for non-stationary
conductors in changing
magnetic fields.
A conducting disk with radius R
lies in the xy-plane and rotates
with constant angular velocity w
about the z-axis. The disk is in a
uniform, constant B field parallel to
the z-axis. Find the induced emf
between the center and the rim of
the disk.
B
Induced electric fields
G
I, dI/dt
If the area vector A points in
the same direction as B set
up by the solenoid, then
 B  BA  0 nIA
d B
dI
 
   0 nA
dt
dt
What force makes the
charges move around the
loop?
d B
  E  dl  
dt
Maxwell’s equations:
Qenclosed
 E  dA 
 B  dA  0
0
d E 

 B  dl  0  ic   0 dt 
d B
E

d
l



dt
Displacement current
Displacement current:
iD  
 B  dl  0 I enclosed
q  CV 
A
Ed  Ed   E
d
dq
d E
iC 

dt
dt
Conduction current
d E
dt
Generalized Ampere’s law:
d E 

B

d
l


i



0 c
0

dt 

Electromagnetic properties
of superconductors
Kammerlingh
Onnes (1911)
discovered
superconductivity.
The critical
temperature for
superconductors is the
temperature at which
the electrical
resistivity of a metal
drops to zero.
Type 1 semiconductors:
Mat. Tc
Be
0
Rh
0
W
0.0
15
Ir
0.1
Lu
0.1
Hf
0.1
Ru
0.5
Os
0.7
Mo
0.9
2
Type 2 semiconductors:
Ma
Zr
Tc
0.546
Mat.
Tc
Al
1.2
Cd
0.56
Pa
1.4
U
0.2
Th
1.4
Ti
0.39
Re
1.4
Zn
0.85
Tl
2.39
Ga
1.083
In
3.40
8
Sn
3.72
2
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The Meissner Effect
“Magnetism and
superconductivity are natural
enemies”. - Lindenfeld
Mixed-State Meissner Effect
Macroscopic magnetization
depends upon aligning the
electron spins parallel to
one another, while
superconductivity depends
upon pairs of electrons with
their spins antiparallel.
Movies:
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/maglev2.html
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/maglev.html#c1
The Eddy currents
The Eddy currents are
induced currents due to
masses of metal moving in
magnetic fields or located in
changing magnetic fields.
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
A “proper” complete circuit is not
necessary for currents caused by
induced emf’s to flow. Microscopic
currents can flow within conductors
and they are eddy currents!
Some applications of that make use of eddy currents:
1. Electromagnetic damping: eddy currents flow in such a
way as to oppose the motion that causes them, acting like
a brake on a moving body.
2. Induction heating: current flows cause heating effects
and eddy currents are no different.
Transformers
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Inductance
Inductance is typified by the behavior of a coil of wire in resisting any change of
electric current through the coil.
A changing current in a coil induces an emf in that same coil!
The coil is the inductor and the relationship between the current
and the emf is described by inductance (self-inductance).
emf   L
dI
dt
When there are 2 or more inductors present, the coupling between the coils is
described by their mutual inductance.
A coil is a reactionary
device, not liking any
change! The induced voltage
will cause a current to flow in
the secondary coil which
tries to maintain the
magnetic field which was
there.
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Mutual inductance
The induced emf in coil 1 is due to self inductance L.
The induced emf in coil 2 is caused by the change in the
current I1:
 2
emf2   N 2
t
Define: N2 B 2  M 21i1
B1
i
  M 21 1
t
t
d 2
di
 N2
 M 21 1
dt
dt

 M 21  N 2 B 2
i1
 emf 2   N 2 A
Show that M12=M21.
Unit of inductance: 1H=1Wb/A=1J/A2
B field from
coil 1 passing
through coil 2
Self-inductance and
Consider a single isolatedinductors
coil. When a varying current is present in a
circuit, it sets up a changing magnetic field that causes a changing magnetic
flux through the same circuit. The resulting emf is called a self-induced emf.
An inductor (or a choke) is a devise
that opposes any current variations
throughout the circuit and is designed
to have a particular inductance.
Inductor:
Define: L 
N B
i
Self induced emf:
(Self inductance)
  L
di
dt
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
A general purpose RF chokes
suitable for power decoupling in
logic circuits, IF tuned circuit
applications and filters etc.
Offers high resonance frequency;
suitable for RF blocking and filtering,
interference suppression in small size
equipment, decoupling and telecoms
and entertainment electronics
RF chokes consisting of a ferrite
based coil former encapsulated in a
polypropylene outer case.
Wire ended toroidal RFI
suppression chokes designed for
use with phase angle control
equipment applications operating
at 240V ac.
http://www.rsphilippines.com/
Magnetic field
di
P  V i energy
Li
a
ab
Variable
source
of emf
L
dt
Pdt  dU  Lidi
1
b
U  Li 2
2
What is the potential difference between
points a and b?
Does energy flow into a resistor whenever a
steady current passes through it?
Does energy flow into a resistor whenever a
varying current passes through it?
Does energy flow into an ideal zero-resistance
inductor when a steady current through it?
Does energy flow into an ideal zero-resistance
inductor when an increasing current passes
through it?
Explain what happens to the bulb
when the switch is closed.
Show that the self-inductance of an ideal
toroidal solenoid of mean radius r and crosssectional area A is:
0 N 2 A
L
2r
Show that the energy density of
an ideal toroidal solenoid is
U
1
N 2i 2
u
 0
2rA 2 2r 2
The RL
circuit
Show that for an ideal toroidal solenoid
B2
u
20
B2
u
2
(magnetic energy
density in a vacuum)
(magnetic energy
density in a material)
In steady state condition, what is the
potential difference between the ends
of the inductor?
The magnetic field H is defined as
B
H
0
Again, an inductor in a circuit makes it
difficult for rapid changes in current to
occur!
Suppose that the switch is initially open,
but is suddenly closed at t=0.
di
V  iR  L
dt
http://farside.ph.utexas.edu/teaching/302l/lectures/node88.html
t R
di '


0 V 0 L dt '
i '
R
i
R
 t
V
 i  1  e L 
R

The time constant for an RL circuit is L/R.
V  iR  L
di
0
dt
The current in a
coil can not
increase (or
decrease) much
faster than L/R.
What is the rate of change of the
current at t=0?
What is the current at the steady
state condition?
V  iR  L
di
0
dt
di
R
  dt
V
L
i
R
Note that Vs/R=I0
Current decay in an RL circuit
 iR  L
i  I 0e
di
0
dt
R
 t
L
http://farside.ph.utexas.edu/teaching/302l/lectures/node88.html
 i 2 R  Li
Vs/R=I0
di
0
dt
When an RL circuit is decaying, what is
the expression of the energy stored in the
inductor as a function of time?
Suppose you want to send a square
wave down a wire. How does the output
signal look like?
The LC circuit
In terms of energy
considerations,
Vi  i 2 R  Li
di
0
dt
Consider a charged capacitor that
is connected to an inductor.
Assume no resistance and no
energy losses to radiation. What
happens to the current in the
circuit?
http://www.sweethaven.com/sweethaven/ModElec/acee/lessonMain.asp?iNum=0402
Kirchoff’s voltage law:
L
di q
 0
dt C
d 2q 1

q0
dt 2 LC
What is the current in the circuit?
What is the stored potential energy
in the capacitor?
What is the stored potential energy
in the inductor?
A solution to the 2nd order differential
equation is:
q  Q coswt  f 
The current is
i  wQ sin wt  f 
What is the physical significance of the
proposed solution?
Are there any possible solutions? What are
they?
Can you consider an LC circuit as a
conservative system?
http://www.greenandwhite.net/~chbut/LC_Oscillator/LC_Oscillator.swf
In terms of energy considerations,
2
2
1 2 q
Q
Li 

2
2C 2C
1
i
Q2  q2
Lc
The RLC circuit
Similarities between SHM and LC circuit
1
1 2
KE  mv 2
ME

Li
2
22
1 2
1
q
PE  kx
EE 
2
2C
2
2
1 2 1 2 1 2
1
1
q
1
Q
2
mv  kx  kA
Li 

2
2
2
2
2C 2 C
v
dx
k

A2  x 2
dt
m
k
w
m
x  A coswt  f 
i
w
dq
1

Q2  q2
dt
Lc
1
LC
q  Q coswt  f 
Kirchoff’s voltage law:
di q
 0
dt C
di 1 t
 iR  L   idt '  0
dt C 0
d 2 q R dq 1


q0
2
dt
L dt LC
 iR  L
Auxiliary equation:
m2 
R
1
m
0
L
LC
2
R 1 R
 1 
m


4
 


2L 2  L 
 LC 
2
2
R 1 R
 1 
m1  

   4

2L 2  L 
LC


R 1 R
 1 
m2  

   4

2L 2  L 
LC


General solution:
2
q  Ae
R
t R
 1 
 t     4

2L 2  L 
 LC 
2
 Be
R t R
 1 
 t     4

2L 2  L 
 LC 
When R2<<4/LC, (underdamped case),
2
R ti  1   R  
  R t  ti 4 1  R 2
 t
4
  

2 L 2  LC   L 
2 L 2  LC   L  
q  Re  Ae
 Be





2
ti  1   R 
  R t ti 4 1  R 2
R

4





 t

q  Re  Ae 2 L e 2  LC   L   Be 2 L e 2  LC   L 


ei  cos   i sin 
Euler’s equation:
i
e
t
 1  R
4
  
2  LC   L 
2





2
2
t
t
 1   R  
 1   R  


 cos
4

 i sin
4

 2  LC   L  
 2  LC   L  




2
t

1
R





q  Ae
cos
4
  f 

 2  LC   L 



R
2


 t
1
R
2L

q  Ae
cos t
 2 f 
w 
 LC 4 L




R
t
2L
2
q  Ae
R
t R
 1 
 t     4

2L 2  L 
 LC 
2
1
R2
 2
LC 4 L
2
 Be
R t R
 1 
 t     4

2L 2  L 
 LC 
R
 1 

4

0
Overdamped:  
 L  2  LC 
R
1 
Critical damping:    4
0
L
 LC 
2
R
 1 

4



0
Underdamped:
L
 LC 
Overdamped
Critically damped
Underdamped
Alternating current
Symbol:
~
Phasors are rotating vectors.
Rectified average value of
a sinusoidal current:
I rav 
I
wt
i  I coswt 
How do we measure
sinusoidally varying current?
2

I max
Root-mean-square current:
-Square the instantaneous current
-Take the average of the sum of the
squares
-Take the square root
i  I coswt 
i 2  I 2 cos 2 wt 
But
1
cos 2 wt   1  cos 2wt 
2
1


i 2  I 2   cos 2wt 
2

http://www.allaboutcircuits.com/vol_3/chpt_3/4.html
cos 2wt  0
I
I rms 
2
Note that I is the maximum current!
Vrms
Phasor diagram:
V

2
The normal voltage source from
local outlets is 220VAC. Is this Vrms?
The current and voltage are in phase!
What are the maximum and minimum
voltages that a TV can have if it is
rated at 110VAC? 110VDC?
Resistance and
reactance
I sin wt
Inductor in an AC circuit:
Resistor in an AC circuit:
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
I sin wt
Phasor diagram:
Note: The phase of the
voltage is defined relative to
the current!
For a pure resistor, the phase is 0. For a pure
inductor, the phase is 90o.
di
dt
d
V   L I sin wt
dt
V  IwL cos wt
  L

V  IwL sin wt  90
The inductive reactance is defined as
X L  wL
The voltage difference between the inductor:
o
V  IX L

Capacitor in an AC circuit:
The voltage leads the current by 90o in phase!
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
I sin wt
The capacitive reactance is defined as
1
XC 
wC
dq
 I sin wt
dt
i
q
I
w
The voltage difference between the capacitor:
cos wt
V
q I

cos wt
C Cw
V
I
sin wt  90 o
Cw

Phasor diagram:
V  IX C
XL

R
XC
The voltage lags the current by 90o in phase!
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Mnemonic for the phase relations of current and
voltage: ELI the ICE man!
VL  IX L
VL  IX L
VR  IX R

VR  IX R
VL  VC
VC  IX C
VC  IX C
V  VR2  VL  VC 
VR  IZ

VL  VC
f
2
VR  IX R
V  I R2  X L  X C 
2
Define:
Z  R2  X L  X C 
2
(Impedance)
The impedance is the ratio of the voltage
amplitude across the circuit to the current
amplitude in the circuit.
For an RLC circuit:

 1 
Z  R 2  wL  

w
C



2
The angle f is the phase angle of the
source with respect to the current.
tan f 
VL  VC

VR
1
wC
R
wL 
At resonance, the phase becomes 0!
Thus,
0  tan
w
1
wC
R
wL 
1
LC
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
The power factor: cos f
For any sinusoidally varying
quantity, the rms value is always
0.707 times the amplitude:
V
I

Z
2
2
Vrms  I rms Z
Power in AC circuits
The instantaneous power delivered to a
circuit element is
A low power factor (large angle f
of lag or lead) means that for a
given potential difference, a large
current is needed to supply a
given amount of power.
-high I2R losses in transmission
-To correct: connect a capacitor
in parallel with the load. WHY?
Transformers:
p  vi
p  V coswt  f I coswt 
p  VI cos f cos 2 wt  VI sin f cos wt sin wt
1
Pave  VI cos f
2
Pave  Vrms I rms cos f
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Derive and expression for Vout/Vin
as a function of the angular
frequency w of the source.
In an LRC series circuit, the
magnitude of the phase angle is
54o, with the source voltage
lagging the current. The
reactance of the capacitor is
350W and the resistor resistance
is 180W. The average power
delivered by the source is 140W.
Find the reactance of the
inductor, the rms current and the
rms voltage.
Derive and expression for Vout/Vin
as a function of the angular
frequency w of the source.
C
~
R
Vout
L
In the circuit shown below, switch S is
closed at time t=0. Find the reading of
each meter just after S is closed.
What does each meter read long after
S is closed?
40W
S
5W
10W
C
~
15W
25V
20mH
10mH
A1
A2
A3
L
R
Vout
A4
Electromagnetic waves
Maxwell’s equations:
Qenclosed
 E  dA   0
 B  dA  0
d E 

 B  dl  0  ic   0 dt 
d B
 E  dl   dt
Speed of light ≡ 299,792,458 m/s
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html