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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
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The magnetic field lines of a current loop
Soleniod Bar magnet
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The bar magnetThe 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
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
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 current-carrying 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.
AB d
dA
dB
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.htmlMagnetic flux density
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 y-coordinate for each ion at any time t is approximately
provided y is remains much smaller than x. • Can this be used for isotope separation?
2
1
2
8
mV
qBxy
Magnetic force on a current-carrying conductor
BlF I
lBAnqvBqvnAlF dd The force on all of the moving charges:
vd
A
l
but dnqvJ IlBF
In general
BlF Idd differential form
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
TorqueBμτ
Where is the direction of the solenoid’s tendency of rotation?
BI
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Sources of magnetic field
20
sin
4 r
vqB
20 ˆ
4 r
q rvB
• The magnetic field produced by a moving charge is proportional to the
– charge
– velocity of the charge
– inverse of the square of the distance
r̂
vqfield point
rsource point
170 104 TmA (permeability constant)
Magnetic field of a current element
r̂
2
0 ˆ
4 r
Id rlB
r
20
sin
4 r
AdlvqndB d
20 sin
4 r
IdldB
nqAdldQ
I
20 ˆ
4 r
Idd
rlB
Biot-Savart law
Bd
field point
source pointld
• For an infinitely long straight wire, the magnetic field at a distance x from the wire is given by
x
I
2
0B
• Ampere’s lawId 0 lB
Force between parallel conductors
r
I
2
0B
x x
Br
lIIlBIF
2
'' 0 I
I’
r
II
l
F
2
'0One 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.
r
Magnetic field of a circular current loop
20 ˆ
4 r
Idd
rlB
I
dl
x
y
z
dBz
dBy
r̂
Let a be the radius of the ring.
222
0
4 ax
dlIdB
sin
4 220
ax
dlIdBy
cos
4 220
ax
dlIdBz
dl
ax
a
ax
IBy
2
122
220 1
4
23
22
20
2 ax
IaBy
What is B at the center of the ring?
If there are N rings, what is B at the center of the rings?
.B
dl Irr
IdlBBdld 0
0 22
lB
The magnetic field at a distance r from a conductor has a magnitude
r
IB
2
0
.B
dl
IdlBdlBd 01 lB
Id 0 lB
.
1
2
3
4
a
d
d
c
c
b
b
addddd lBlBlBlBlB
ab
c
d
a
d
d
c
c
b
b
adlBdldlBdld 13 00lB
a
d
d
c
c
b
b
adlBdldlBdld 13 00lB
r1
r2
022
22 2
2
01
1
0
rr
Ir
r
Id
lB
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
enclosedId 0 lBx
.
.x
x
. x path of integration
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
kJ ˆ12
2
20
a
r
a
I
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
0J
for ra
for ra
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 ra.
c) Obtain and expression for the current I contained in the circular cross section of radius ra 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 ra.
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.
.
x
a
ax
P
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?
I2R
D
I1
Study the examples in the book!
Field inside a long cylindrical conductor
I
Amperian loop
Rr
enclosedId 0 lB
2
2
02R
rIrB
B
20
2 R
rIB
r
B
R
Field of a solenoid
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
nLIBL 0enclosedId 0 lB
Let n be the number of turns.
nIB 0
Find the magnetic field of a toroidal solenoid.
Magnetic materials
e
I
L
IA
r
ev
T
eI
2
22
2 evrr
r
ev
But vrL mmvrL
Lm
e
2
The dipole moment’s component in a particular direction is an integral multiple of
2
h
Js10626.6 34hWhen 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.
2
hL If 224 Am10274.9
4
m
eh
The Bohr magneton
Vi
i
μM
ParamagnetismThe magnetization of the material is proportional to the applied magnetic field in which the material is placed.
Magnetization
I
MμBB 00
Curie’s law
TC
BM Mmagnetization Ttemperature
(K)
CCurie’s constant Bmagnetic field
For a given ferromagnetic material the long range order abruptly disappears at a certain temperature called the Curie temperature.
MaterialCurie 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 578Data 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)
The relative permeability:
0 mK
For common paramagnetic materials, Km varies from 1.00001 to 1.003.
The magnetic susceptibility:
1 mm KThe magnetic susceptibility is the amount by which the relative permeability differs from unity.
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 -1.0Young &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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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.
Average magnetic dipole
moment per atom of iron
Electromagnetic induction
• Results from experiments– 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.
galvanometer
N
S
electromagnet
Electromagnetic induction
• Results from experiments– 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.
galvanometer
N
S
electromagnet
Electromagnetic induction
• Results from experiments– 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.
galvanometer
N
S
electromagnet
Electromagnetic induction• Results from experiments
– 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.
galvanometer
N
S
electromagnet
Electromagnetic induction
• Result from experiments– 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.
What is common in these results?
galvanometer
N
S
electromagnet
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.
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.
Electromagnetic induction
dt
d B
AB dB
dt
dN B
For N turns,
Direction of induced emf
• 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 dB/dt is positive, then the induced emf or current is negative.
• If the flux is decreasing, dB/dt is negative and the induced emf or current is positive.
AIncreasingB
<0
ADecreasing B
>0
Direction of induced emf
• Determine the direction of the induced emf or current using your right hand. Curl the fingers of your right hand around the A vector, with your right thumb in the direction of A. If the induced emf or current in the circuit is positive, it is in the same direction as your curled fingers. If the induced emf or current is negative, it is in the opposite direction.
A
IncreasingB
>0
ADecreasing B
<0
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.
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
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
ExamplesTwo 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 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.
R
S a b
An alternator is a device that generates an emf. A rectangular loop is made to rotate with a constant angular velocity about the axis shown below. The magnetic field B is uniform and constant. At time t=0, =0, determine the induced emf.
A B
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 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
ExamplesConsider a U-shaped conductor in a uniform B-field. If a metal with length L is put across the arms of the conductor, forming a circuit, and move the rod to the right with constant velocity v, find the magnitude and direction of the resulting emf.
vxx
xx
x x xxx x
xx
xxx
x x x x x
xx
x
xx
xxxx
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) the current is from b to a and increasing? a b Winding A
Winding B
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?
Examples
v v
I I
Motional electromotive force
v
b
ax 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 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 x x x x x x x x x x x x x
I
vBLMotional electromotive force:
L
LBv dd
LBv dThis equation can be used for non-stationary conductors in changing magnetic fields.
B
A conducting disk with radius R lies in the xy-plane and rotates with constant angular velocity 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.
Induced electric fields
G
If the area vector A points in the same direction as B set up by the solenoid, then
nIABAB 0
dt
dInA
dt
d B0
I, dI/dt
What force makes the charges move around the loop?
Maxwell’s equations:
0enclosedQ
d AE
0 AB d
dt
did Ec 00 lB
dt
dd B
lE
dt
dd B
lE
Displacement current
enclosedId 0 lB
EEdEdd
ACVq
dt
d
dt
dqi EC
Conduction current
Displacement current:
dt
di ED
dt
did Ec 00 lB
Generalized Ampere’s law:
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 0.015
Ir 0.1
Lu 0.1
Hf 0.1
Ru 0.5
Os 0.7
Mo 0.92
Mat. Tc
Al 1.2
Pa 1.4
Th 1.4
Re 1.4
Tl 2.39
In 3.408
Sn 3.722
MaZr
Tc0.546
Cd 0.56
U 0.2
Ti 0.39
Zn 0.85
Ga 1.083
Type 2 semiconductors:
http://www.physnet.uni-hamburg.de/home/vms/reimer/htc/pt3.html
The Meissner Effect
Mixed-State Meissner Effect
“Magnetism and superconductivity are natural enemies”. - Lindenfeld
Macroscopic magnetization depends upon aligning the electron spins parallel to one another, while superconductivity depends upon pairs of electrons with their spins antiparallel.
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/maglev.html#c1
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/maglev2.html
Movies:
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
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http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
InductanceInductance 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!
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
The coil is the inductor and the relationship between the current and the emf is described by inductance (self-inductance).
dt
dILemf
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.
Mutual inductanceThe 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:
t
iM
t
BANemf
121
122
tNemf
222
dt
diM
dt
dN 1
212
2
1
2221 i
NM B
Show that M12=M21.
Unit of inductance: 1H=1Wb/A=1J/A2
12122 iMN B Define:B field from coil 1 passing through coil 2
Self-inductance and inductors
An inductor (or a choke) is a devise that opposes any current variations throughout the circuit and is designed to have a particular inductance.
i
NL B (Self inductance)
dt
diLSelf induced emf:
Inductor:
Consider a single isolated 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.
Define:
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http://www.rsphilippines.com/
Wire ended toroidal RFI suppression chokes designed for use with phase angle control equipment applications operating at 240V ac.
RF chokes consisting of a ferrite based coil former encapsulated in a polypropylene outer case.
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
Explain what happens to the bulb when the switch is closed.
Variable source of emf
L
a
bWhat is the potential difference between points a and b?
Magnetic field energy
dt
diLiiVP ab
LididUPdt
2
2
1LiU
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?
Show that the self-inductance of an ideal toroidal solenoid of mean radius r and cross-sectional area A is:
r
ANL
2
20
The RL circuit
Show that the energy density of an ideal toroidal solenoid is
222
022
1
2 r
iN
rA
Uu
Show that for an ideal toroidal solenoid
0
2
2B
u
The magnetic field H is defined as
0B
H
(magnetic energy density in a vacuum)
(magnetic energy density in a material)2
2Bu
Again, an inductor in a circuit makes it difficult for rapid changes in current to occur!
In steady state condition, what is the potential difference between the ends of the inductor?
http://farside.ph.utexas.edu/teaching/302l/lectures/node88.html
dt
diLiRV
Suppose that the switch is initially open, but is suddenly closed at t=0.
0dt
diLiRV
0dt
diLiRV
What is the rate of change of the current at t=0?
What is the current at the steady state condition?
dtL
R
RV
i
di
tidt
L
R
RV
i
di00
''
'
tL
R
eR
Vi 1
The time constant for an RL circuit is L/R.
http://farside.ph.utexas.edu/teaching/302l/lectures/node88.html
Current decay in an RL circuit
0dt
diLiR
tL
R
eIi
0
The current in a coil can not increase (or decrease) much faster than L/R.
Note that Vs/R=I0
Suppose you want to send a square wave down a wire. How does the output signal look like?
Vs/R=I0
http://www.sweethaven.com/sweethaven/ModElec/acee/lessonMain.asp?iNum=0402
In terms of energy considerations,
02 dt
diLiRiVi
02 dt
diLiRi
When an RL circuit is decaying, what is the expression of the energy stored in the inductor as a function of time?
The LC circuit
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?
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?
http://www.greenandwhite.net/~chbut/LC_Oscillator/LC_Oscillator.swf
0C
q
dt
diL
Kirchoff’s voltage law:
01
2
2
qLCdt
qd
A solution to the 2nd order differential equation is:
tQq cos
The current is
tQi sin
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?
In terms of energy considerations,
C
Q
C
qLi
222
1 222
221qQ
Lci
The RLC circuit
Kirchoff’s voltage law:
0C
q
dt
diLiR
01
2
2
qLCdt
dq
L
R
dt
qd
0'1
0
tidt
Cdt
diLiR
Similarities between SHM and LC circuit
2
2
1mvKE 2
2
1LiME
2
2
1kxPE
C
qEE
2
2
1
222
2
1
2
1
2
1kAkxmv
C
Q
C
qLi
222
2
1
2
1
2
1
22 xAm
k
dt
dxv 221
qQLcdt
dqi
m
k
LC
1
tAx cos tQq cos
012 LC
mL
Rm
Auxiliary equation:
LCL
R
L
Rm
14
2
1
2
2
LCL
R
L
Rm
14
2
1
2
2
1
LCL
R
L
Rm
14
2
1
2
2
2
LCL
Rtt
L
R
LCL
Rtt
L
R
BeAeq1
422
14
22
22
General solution:
When R2<<4/LC, (underdamped case),
221
422
14
22Re L
R
LC
tit
L
R
L
R
LC
tit
L
R
BeAeq
22
14
22
14
22Re L
R
LC
tit
L
RL
R
LC
tit
L
R
eBeeAeq
Euler’s equation: sincos iei
221
42 1
42
sin1
42
cos
2
L
R
LC
ti
L
R
LC
te L
R
LC
ti
2
2 14
2cos
L
R
LC
tAeq
tL
R
2
22
4
1cos
L
R
LCtAeq
tL
R
2
2
4
1
L
R
LC
Overdamped:
LCL
Rtt
L
R
LCL
Rtt
L
R
BeAeq1
422
14
22
22
01
42
LCL
R
Critical damping: 01
42
LCL
R
Underdamped: 01
42
LCL
R
Overdamped
Critically damped
Underdamped
Alternating current~Symbol:
Phasors are rotating vectors.
tIi cos
I
t
How do we measure sinusoidally varying current?
http://www.allaboutcircuits.com/vol_3/chpt_3/4.html
max
2II rav
Rectified average value of a sinusoidal current:
Root-mean-square current:-Square the instantaneous current-Take the average of the sum of the squares-Take the square root
tIi cos
tIi 222 cosBut
tt 2cos12
1cos2
tIi 2cos
2
122
2
II rms
02cos t
Note that I is the maximum current!
2
VVrms
The normal voltage source from local outlets is 220VAC. Is this Vrms?
What are the maximum and minimum voltages that a TV can have if it is rated at 110VAC? 110VDC?
Resistance and reactance
Resistor in an AC circuit:
Phasor diagram:
The current and voltage are in phase!
Inductor in an AC circuit:tI sin
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Phasor diagram:
The voltage leads the current by 90o in phase!
dt
diL
tIdt
dLV sin
tLIV cos
tI sin
otLIV 90sin
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.
The inductive reactance is defined as
LX L
LIXV The voltage difference between the inductor:
Capacitor in an AC circuit:
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
tIdt
dqi sin
tI
q
cos
tI sin
tC
I
C
qV
cos
otC
IV 90sin
Phasor diagram:
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.htmlThe voltage lags the current by 90o in phase!
The capacitive reactance is defined as
CXC
1
The voltage difference between the capacitor:
CIXV
R
LX
CX
Mnemonic for the phase relations of current and voltage: ELI the ICE man!
CC IXV
LL IXV RR IXV
CC IXV
LL IXV RR IXV
CL VV
RR IXV
CL VV
IZVR
22CLR VVVV
22CL XXRIV
Define: 22CL XXRZ (Impedance)
The impedance is the ratio of the voltage amplitude across the circuit to the current amplitude in the circuit.
For an RLC circuit:
2
2 1
C
LRZ
The angle is the phase angle of the source with respect to the current.
RC
L
V
VV
R
CL
1
tan
At resonance, the phase becomes 0! Thus,
LC
1
RC
L
1
tan0
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Power in AC circuits
For any sinusoidally varying quantity, the rms value is always 0.707 times the amplitude:
ZIV
22
ZIV rmsrms
The instantaneous power delivered to a circuit element is
vip
tItVp coscos
ttVItVIp sincossincoscos 2
cos2
1VIPave
cosrmsrmsave IVP
The power factor: cos
A low power factor (large angle 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:
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 of the source.
R L
C
~
Vout
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 350and the resistor resistance is 180. 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 of the source.
R L
C~ Vout
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?
40
5
20mH
1015
A2 A3 A4A1
10mH
S
25V
Electromagnetic waves
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Speed of light ≡ 299,792,458 m/s
Maxwell’s equations:
0enclosedQ
d AE
0 AB d
dt
did Ec 00 lB
dt
dd B
lE