Exam 3 Lectures Magnetism. Definitions Magnetic field—The vector field a magnet produces all...
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Transcript of Exam 3 Lectures Magnetism. Definitions Magnetic field—The vector field a magnet produces all...
Exam 3 Lectures
Magnetism
Definitions• Magnetic field—The vector field a magnet
produces all around itself to interact with its environment (produces force)
• Tesla—the unit of the magnetic field
• Permanent magnets—have a permanent magnetic field without outside influences
• Electromagnets—magnets, which have a magnetic field in the presence of a current but not without it
AmN
msCN
smC
NT
1111
More Definitions
• Monopoles—magnetic charges (not found in nature)
• North pole—end from which field lines emerge
• South pole—end where field lines enter.• Crossed fields—an E field and B field
present that are perpendicular to each other
Magnetic Fields• Produced by:
1. Moving electrically charged particles
2. Elementary particles (such as electrons) have an intrinsic magnetic field around them—basic characteristic of such particles
• On the earth the south magnetic pole is close to the north geographic pole, and the north magnetic pole is close to the south geographic pole
• A C shaped magnet is used to get a uniform magnetic field in experiments
• Opposite magnetic poles attract each other
• Like magnetic poles repel each other
Magnetic Field Lines and Magnetic Fields• The direction of the magnetic B field at
any point on a B field line is in the direction of the tangent to the B field line.
• The spacing of the lines represents the magnitude of the B field. The B field is stronger where the B field lines are closer together.
• The B field lines all pass through the magnet forming closed loops: they go in one side and out the other
• Force a moving charged particle feels from an external magnetic field
• Must use the right hand rule for direction
qvBBqvqvBF
BvqF
B
B
sin
Differences Between Electric and Magnetic Forces
• The electric force acts in the direction of the E field, whereas the magnetic force acts perpendicular to the B field.
• The electric force acts on a charged particle whether or not it is moving, whereas the magnetic force acts on a charged particle only if it is moving.
• The electric force does work in displacing a charged particle, whereas the magnetic force does no work when a charged particle is displaced
Circulating Charged Particles• Important that force is perpendicular to
velocity; therefore force can change direction but not magnitude of velocity
• Particles under the influence of the magnetic force undergo uniform circular motion
qBmv
qvBmv
r
rv
mqvBF
2
2qBm
qBmv
vvr
T 222
mqB
Tf
21
mqB
f 2
Circulating Charged Particles cont• If velocity of charged particle has
component parallel to magnetic field, the particle will move in a helical path
• Parallel component – pitch of helix
• Perpendicular component – radius of helix
cos|| vv
sinvv
Circulating charges have uses and examples in nature:
1. The magnetic bottles used in some experiments2. Cyclotrons3. Synchrotrons
The van Allen belts of the earth
Current Carrying Wire• Current consists of charges moving
along the wire
B d
d
F q v B
qv B nAL
iL B
BsidFd B
baB BsdiF
same direction of force if we assume positive or negative charge carriers
Torque
Torque tends to rotate the loop so that A is rotated into the direction of B
BBAi
iABiabB
ibBa
ibBa
Frnet
sinsin
sin2
sin2
Potential EnergyElectric Magnetic
sinpEEp
sinBB
BANi
cospEEpU cosBBU
Crossed Fields – Hall Effect1. Charged particles moving through the conductor
subject to an external B field – get a magnetic force acting on them.
2. Magnetic force deflects the charged particles in the direction of the force and makes them line up on one wall of the conductor.
3. One wall of the conductor is more negative and the other is more positive charge – electric field is set up.
4. Eventually electric force balances magnetic force and the charges are allowed to continue on their way.
5. One wall will be at a higher potential than the other.
6. By looking at the potential difference between the two walls the sign the charges may be determined
a) Shows the situation for which the charge carriers are negative
b) Shows the situation for which the charge carriers are positive
Mass Spectrometer• Another use of crossed fields
mqV
vmvqV2
21 2
2
22
qB
VmmqV
qBm
qBmv
r
VBr
qm
VBqr
m
2
222
22
Calculating Fields Comparison
Electric Magnetic Symmetry
No symmetry
symmetry
rr
dqkE e ˆ2
o
encqAdE
2
ˆ4 r
rsdIB o
inoIdsB
Calculating Magnetic Field due to a Current• Magnetic fields are produced by moving
charges (currents)
• Biot Savart Law
2
ˆ4 r
rsdIB o
AmpTm
o7104
Some Magnetic Fields
• Long Straight Wire
• Half a Long Straight Wire
• Circular Arc of Wire
Ri
B2
0
Ri
B4
0
Ri
B
40
Force Between 2 Parallel Currents
• Parallel currents attract, antiparallel currents repeldii
Ldi
LiBLiF aboaobabab
22
Ampere’s Law
• This is a line integral to be integrated around a closed loop (Amperian loop)
• To use Ampere’s Law1. First decide which type of symmetry best
complements the problem
2. Draw an Amperian loop (mathematical not real) reflecting the symmetry you chose around the current distribution through the point of interest.
inoIdsB
Long Cylindrical Conductor
INSIDElinear
OUTSIDEhyperbolic
Solenoid• Solenoid – a long tightly wound helical coil
of wire
ad
dc
cb
ba
sdBsdB
sdBsdBsdB
inB 0
Toroid• Toroid – a solenoid bent into a doughnut
shape
0
2iN
Br
Definitions• Induced current—current produced in a loop by a
changing magnetic field• Induced emf—work done per unit charge in producing
a current.• Induction—process of producing current and emf• Faraday’s law of induction—an emf is induced in a
loop when the number of magnetic field lines that pass through the loop changes.
• Lenz’s law—an induced current has a direction such that the magnetic field due to the current opposes the change in magnetic field that induces the current. The direction of the induced emf is the direction of the induced current
First of Two Experiments
• Three discoveries:1. Current appears only if there is relative
motion between the loop and magnet
2. Faster motion between the loop and magnet produces greater current
3. Opposite motion of the magnet produces opposite direction of current
Second of Two Experiments
• Current in a wire produces a magnet field• As the current increases the magnetic field
increases• As the magnetic field increases a current
appears.
Magnetic Flux
• Units of magnetic flux is the Weber
• Three terms and therefore three ways that flux can change with time
coscos BABdAAdBB
dtd
dtdA
dtdB cos
,,
211 TmWb
Induction – Faraday’s Law• Faraday’s Law – an emf is induced in a loop
when the number of magnetic field lines passing through the loop changes
• The negative only tells us direction and we will ignore it (unless I have a reason not to)
• The number of field lines doesn’t matter, just the rate of change of the number of field lines determines the induced emf and current
cosBAdtd
AdBdtd
dtd B
Induction cont• Notice there are three terms and therefore
three ways that flux can change: 1. The magnitude of the magnetic field can
change
2. The area of the coil can change
3. The angle between the direction of the magnetic field and the coil can change
dtdB
NAdtd
N B cos
dtdA
NBdtd
N B cos
dtd
NBAdtd
N B cos
Lenz’s Law• An induced current has a direction such that the
magnetic field due to the current opposes the change in the magnetic flux that induces the current
• The direction of the induced emf is the direction of the induced current
Solving Problems1. First know the direction of the external changing B
field
2. Next note how the external B field is changing
3. Use the two rules below to determine the direction the induced B field must have
4. Test with to determine the direction of the induced current to give the appropriate direction of the induced B field
5. It must follow the two rules below:a) If the external B field is increasing, the induced B field is
in the opposite direction of the external B field
b) If the external B field is decreasing, the induced B field is in the same direction of the external B field
rsd ˆ
Energy Transfer• During induction thermal
energy is produced by the work done to the system
RBLv
Ri
BLvdtdx
BLdtdA
Bdtd
RRvLB
FvP
RvLB
LBRBLv
iLBF
2222
22
Electric Field Induction
a) Ring with current induced
b) No ring but E field induced
c) E field lines of induced E field
d) Four closed paths of identical areas: 1 & 2 have same emf, 3 smaller emf, 4 no emf
dtd
sdE B
E Field Induction cont• The electric field lines for induced
electric fields always produce closed loops
• The electric field lines for static charges start at positive charges and end in negative charges
• The electric potential only has meaning for electric fields produced by static charges—it has no meaning for electric fields produced by induction
dtd
sdEsdE Bfinalinitial
0
Definitions• Inductor—a device used in a circuit to
produce a desired magnetic field, usually a solenoid
• Inductance— where N is the number
of turns, i is the current in windings, and is
the magnetic flux.
• Henry—the unit of inductance
iN
L
ATm
H2
11
More Definitions• Magnetic flux linkage— since N windings
are “linked” by shared flux
• Self-induction—a time varying current in a circuit produces in the circuit an induced emf opposing the emf, which initially set up the time varying current
• Mutual induction—the process by which a coil produces an induced emf in another coil and visa versa, the mutual interaction (induction) of 2 coils
N
Inductance• Distinction between source emf and
induced emf – The source emf and current is caused by a
physical source such as a battery.– The induced emf and current is caused by
a changing magnetic field
• self-induced emf or back emf
NLiiN
L
dtdi
LdtLid
dtNd
L
Inductance of SolenoidnlN
inABA o
AnlL
lAniinAnl
iN
L
o
oo
2
2
•Inductance depends only on the geometry of the device•L is a constant of proportionality between the
magnetic flux and the current in the circuit
Groups of Inductors• Series
• Parallel
• Note we must be careful, this is true only if there is a large distance between the inductors; otherwise we will have a mutual inductance problem
321 LLLLequivalent
321
1111LLLL eequivalenc
Energy Stored in a Magnetic Field
• This holds for all magnetic fields not just solenoids even though we used a solenoid to calculate it
dtdi
LiRii 2
2
21LiU
LididU
B
B
dtdU
timeenergy
dtdi
LiP B
o
ooBB
B
B
inAAin
Ai
lL
Al
Li
AlU
VU
u
2
222222
21
21
2212
1
Self and Mutual Induction• Magnetic flux through coil 2 associated with
a current in coil 1 which links the N2 turns of
coil 2 is
• Mutual inductance M21 of coil 2 wrt coil 1
• M12 = M21 = M
21
Self Induction Mutual Induction
NLiiN
L
dtdi
LL
2121211
21221 NiM
iN
M
dtdi
M 1212
RL Circuits• RL circuit – another circuit in which the
current varies with time
L
t
eR
i 1
L
t
eR
i
Charging Discharging
dtdi
LiR
RL
L
L
t
L e
Magnetic Materials• The simplest magnetic structure that can exist
is a magnetic dipole
• Gauss’s Law for magnetic fields
since there are no magnetic monopoles
• Electrons have two types of magnetic dipole moments:
1. Spin magnetic dipole moment associated with its spin
2. Orbital magnetic dipole moment associated with its orbiting about the nucleus
0AdBB
Magnetism of the Earth• The earth is a huge magnet and has
the magnetic field of a huge dipole• Remember field lines enter in
magnetic north pole; therefore the geomagnetic south pole is near the geographic north pole
• Field declination – the angle (left or right) between geographic north and the horizontal component of the earth’s field
• Field inclination – the angle (up or down) between a horizontal plane and the earth’s field direction
Diamagnetism
• Diamagnetism – (exhibited by all common materials) weak magnetic dipole moments are produced in the atoms of the material placed in an external B field. The combination of all these induced dipole moments give the material only a feeble net B field which disappears when the external B field is removed
Paramagnetism• Paramagnetism – (exhibited by materials
containing transition elements, rare earth elements, and actinide elements) each atom of material has a permanent magnetic dipole moment which is random and the net magnetic field is approximately 0. An external magnetic field may align material moments giving a net magnetic field which disappears when the external B field is removed
Ferromagnetism
• Ferromagnetism – (exhibited by iron, nickel, and certain other elements) some electrons in these materials align their resultant magnetic dipole moments to produce regions with strong magnetic dipole moments. External magnetic fields align these magnetic moments producing a strong B field which partially persists when the external B field is removed
Ferromagnetism
The arrows show the directions of magnetization in the given domains. Domains that are magnetized in the direction of the applied magnetic field grow stronger.
Hysteresis Loops
Induced Magnetic Fields• Faraday’s Law of Induction
• Maxwell’s Law of Induction
• Ampere – Maxwell Law
• Displacement current
dtd
sdE B
dtd
sdB E00
encE i
dtd
sdB 000
dE i
dtd
0
Maxwell’s Equations
Name Integral Form Differential Form
Gauss’s Law
Gauss’s Law
Faraday’s Law
Ampere-Maxwell’s Law
S o
QAdE
S
AdB 0
dtd
sdE B
dtd
isdB Eooo
o
E
0 B
tB
E
tE
JB ooo
Definitions
• Del—
• Gradient—(of a scalar function)
• The gradient of a function points in the direction of maximum increase of the function S. The magnitude of the gradient gives the slope or rate of increase along this maximal direction.
kz
jy
ix
ˆˆˆ
kzS
jyS
ixS
Skz
jy
ix
S ˆˆˆˆˆˆ
More Definitions• Divergence—(of a vector)
• The divergence is a measure of how much the vector spreads out or diverges from the point in question.
zV
y
V
xV
kVjViVkz
jy
ix
V
zyx
zyx
ˆˆˆˆˆˆ
More Definitions• Curl—(of a vector)
• The curl is a measure of how much the vector “curls around” the point in question
kyV
x
Vj
zV
xV
iz
V
yV
VVVzyx
kji
kVjViVkz
jy
ix
V
xyxzyz
zyx
zyx
ˆˆˆ
ˆˆˆ
ˆˆˆˆˆˆ
Conversion from Integral to Differential• Green’s Theorem
• Stoke's Theorem
• For Gauss’ Law
AdVdV
sdVdAV
o
oo
S o
E
VQ
E
QdEAdE
• Remember that we could find the electric field from the potential function
• There is also a magnetic vector potential A, such that
• The Lorentz force law
• Together with the Maxwell’s equations, this force law completely describes all classical electromagnetic interactions
sV
Es VE
AB
BvqEqF
LC Oscillations• LC Circuit – oscillating circuit
LC Oscillations cont
dtdq
Cq
dtdi
LiCqLi
dtd
dtdU
CqLi
UUU EBtotal
22
2222
22
tqq cosmax
maxmax
maxmax sinsin
qi
titqdtdq
i
LC1
C
qtt
Cq
tC
qt
Cq
UU EB
2cossin
2
cos2
sin2
2max22
2max
22max2
2max
RLC Circuit – Damped Oscillating Circuit
0
0
0
2
2
2
2
Cq
dtdq
Rdt
qdL
Cq
iRdtdi
L
RiiCq
dtdi
Li
dtdq
Cq
dtdi
LiRidtdU
teqq LRt
cos2max
2
222
4
12 L
RLCL
R
oscillates but witha decaying amplitude
RLC Circuit cont• For RLC circuits, if we put energy in at
the rate in which it is taken out by the resistor, we will stop the damping of the oscillations (forced oscillations)
• The driving angular frequency is the frequency of the external alternating emf producing the driving force
• Resonance occurs and the amplitude of the current in the circuit is a maximum if
d
CL
RL
RLCL
R 40
4
12 2
222 Criticallydamped
AC Circuits• In AC circuits the power source is
alternating and the voltage and current are sinusoidal
• Power is still lost through resistors even with an alternating power source
maxmax 707.
2i
iirms
RiRiRi
P rmsav2
2max
2max
22
0avei
Resistive Load
VR and iR are in phase
0 RV
titR
VRV
i
tVV
tV
RRR
R
RR
R
sinsin
sin
sin
maxmax
max
max
Capacitive Load
VC and iC are 90° out of phase and iC leads VC
0 CV
90sin
90sin
cos
sin
sin
sin
max
max
max
max
max
max
ti
tX
V
tCVdtdq
i
tCVCVq
tVV
tV
C
C
C
CC
C
CCC
CC
C
CXC
1
Inductive Load
VL and iL are 90 out of phase and VL leads iL
0 LV
LV
dtdi
dtdi
LtVV
tV
LLLLL
L
sin
sin
max
max
90sin90sin
cos
sin
maxmax
max
max
titX
V
tL
Vi
tL
Vdtdi
LL
L
LL
LL
LX L
Series RLC Circuit• Solve vectorally using
phasors
• Z is the impedance and is a resistance
• To get the phase we again use the phasors and get
LCR VVV
ZXXRi
CL
22max
RXX CL tan