Chapter 20: Electromagnetic · PDF filePHY2054: Chapter 20 11 Faraday’s Law of Induction...
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Transcript of Chapter 20: Electromagnetic · PDF filePHY2054: Chapter 20 11 Faraday’s Law of Induction...
PHY2054: Chapter 20 2
TopicsElectromagnetic Induction
Magnetic fluxInduced emf
Faraday’s Law Lenz’s LawMotional emf
Magnetic energyInductanceRL circuitsGenerators and transformers
PHY2054: Chapter 20 3
Reading Quiz 1Magnetic flux through a wire loop depends on:
1) thickness of the wire2) resistivity of the wire3) geometrical layout of the wire4) material that the wire is made of5) none of the above
Flux depends only on geometrical properties cosB BA θΦ =
PHY2054: Chapter 20 4
Reading Quiz 2An induced emf produced in a motionless circuit is due to
1) a static (steady) magnetic field2) a changing magnetic field3) a strong magnetic field4) the Earth’s magnetic field5) a zero magnetic field
Faraday’s law
PHY2054: Chapter 20 5
Reading Quiz 3Motional emf relates to an induced emf in a conductor which is:
1) long2) sad3) stationary4) insulated5) moving Potential difference proportional to velocity
PHY2054: Chapter 20 6
Reading Quiz 4Faraday’s law says that1. an emf is induced in a loop when it moves through an electric
field2. the induced emf produces a current whose magnetic field
opposes the original change3. the induced emf is proportional to the rate of change of magnetic
flux
Faraday’s law
PHY2054: Chapter 20 7
Reading Quiz 5A generator is a device that:
a) transforms mechanical into electrical energyb) transforms electrical into mechanical energyc) transforms low voltage to high voltage
PHY2054: Chapter 20 8
Magnetic FluxDefine magnetic flux ΦB
θ is angle between B and the normal to the plane
Flux units are T-m2 = “webers”
When B field is not constant or area is not flat
Integrate over area
Won’t cover this case here
cosB BA θΦ = ⋅ =B A
B AdΦ = ⋅∫ B A
PHY2054: Chapter 20 10
This effect can be quantified by Faraday’s Law
Experimental Observation of Induction
PHY2054: Chapter 20 11
Faraday’s Law of Induction
inducedemf
number of loops
rate of changeof flux with time
The faster the change, the larger the induced emfFlux change caused by changes in either B, area, orientationMinus sign explained next slide
BNt
ε ΔΦ= −
Δ
PHY2054: Chapter 20 12
Faraday’s Law of Induction
inducedemf
number of loops
rate of changeof flux with time
Minus sign from Lenz’s Law:Induced current produces a magnetic field which opposes the original change in flux
BNt
ε ΔΦ= −
Δ
PHY2054: Chapter 20 13
Comment on Lenz’s LawWhy does the induced current oppose the change in flux?
Consider the alternativeIf the induced current reinforced the change… then the change would get bigger… which would then induce a larger current… and then the change would get even bigger… and so on . . .
A clear violation of conservation of energy!!
PHY2054: Chapter 20 14
Bar magnet moves through coilCurrent induced in coil
Reverse poleInduced current changes sign
Coil moves past fixed bar magnetCurrent induced in coil as in (A)
Bar magnet stationary inside coilNo current induced in coil
Direction of Induced Current
S N
vS N
vN S
S N
A
B
C
D
v
PHY2054: Chapter 20 15
ConcepTest: Lenz’s Law If a North pole moves towards the loop from above the page, in what direction is the induced current?
(a) clockwise (b) counter-clockwise (c) no induced current
Must counter flux change indownward direction with upward B field
PHY2054: Chapter 20 16
ConcepTest: Induced Currents A wire loop is being pulled through a uniform magnetic field. What is the direction of the induced current?
(a) clockwise (b) counter-clockwise (c) no induced current
x x x x x x x x x x x x xx x x x x x x x x x x x xx x x x x x x x x x x x xx x x x x x x x x x x x xx x x x x x x x x x x x xx x x x x x x x x x x x xx x x x x x x x x x x x x
No change in flux, no induced current
PHY2054: Chapter 20 17
ConcepTest: Induced Currents
In the cases above, what is the direction of the induced current?
x x x x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x x
1
2
Counterclockwise None
PHY2054: Chapter 20 18
ConcepTest: Lenz’s Law If a coil is shrinking in a B field pointing into the page, in what direction is the induced current?
(a) clockwise (b) counter-clockwise (c) no induced current
Downward flux is decreasing,so need to create downwardB field
PHY2054: Chapter 20 19
Induced currents A circular loop in the plane of the paper lies in a 3.0 T magnetic field pointing into the paper. The loop’s diameter changes from 100 cm to 60 cm in 0.5 s
What is the magnitude of the average induced emf?What is the direction of the induced current?If the coil resistance is 0.05Ω , what is the average induced current?
Direction = clockwise (Lenz’s law)
Current = 3.016 / 0.05 = 60.3 A
( )2 20.3 0.53.0 3.016 Volts
0.5BV
t
π −ΔΦ= = × =
Δ
PHY2054: Chapter 20 20
ConcepTest: Induced Currents A wire loop is pulled away from a current-carrying wire. What is the direction of the induced current in the loop?
(a) clockwise (b) counter-clockwise (c) no induced current
I
Downward flux through loopdecreases, so need to createdownward field
PHY2054: Chapter 20 21
ConcepTest: Induced Currents A wire loop is moved in the direction of the current. What is the direction of the induced current in the loop?
(a) clockwise (b) counter-clockwise (c) no induced current
I
Flux does not change whenmoved along wire
PHY2054: Chapter 20 22
ConcepTest: Lenz’s Law If the B field pointing out of the page suddenly drops to zero, in what direction is the induced current?
(a) clockwise (b) counter-clockwise (c) no induced current
If a coil is rotated as shown, in a B field pointing to the left, in what direction is the induced current?
(a) clockwise (b) counter-clockwise (c) no induced current
Upward flux through loopdecreases, so need to createupward field
Flux into loop is increasing, soneed to create field out of loop
PHY2054: Chapter 20 23
ConcepTest: Induced CurrentsWire #1 (length L) forms a one-turn loop, and a bar magnet is dropped through. Wire #2 (length 2L) forms a two-turn loop, and the same magnet is dropped through. Compare the magnitude of the induced currents in these two cases.
(a) I1 = 2 I2
(b) I2 = 2 I1
(c) I1 = I2 ≠ 0(d) I1 = I2 = 0(e) Depends on the strength of the magnetic field
Voltage doubles, but R alsodoubles, leaving current the same
PHY2054: Chapter 20 24
Motional EMFConsider a conducting rod moving on metal rails in a uniform magnetic field:
( ) ( )ε Φ= = = =Bd d BA d BLx dxBL
dt dt dt dt
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 xx x x x x x x x x
v L ε = BLv
Current will flow counter-clockwise in this “circuit”. Why?
PHY2054: Chapter 20 25
Force and Motional EMFPull conducting rod out of B field
Current is clockwise. Why?
Current within B field causes force
Force opposes pull (RHR)Also follows from Lenz’s law
We must pull with this forceto maintain constant velocity
ε= =
BLviR R
2 2= =
B L vF iLBR
PHY2054: Chapter 20 26
Power and Motional EMFForce required to pull loop:
Power required to pull loop:
Energy dissipation through resistance
Same as pulling power! So power is dissipated as heatKinetic energy is constant, so energy has to go somewhereRod heats up as you pull it
2 2= =
B L vF iLBR
2 2 2= =
B L vP FvR
2 2 2 22 ⎛ ⎞= = =⎜ ⎟
⎝ ⎠BLv B L vP i R R
R R
PHY2054: Chapter 20 27
ExamplePull a 30cm x 30cm conducting loop of aluminum through a 2T B field at 30cm/sec. Assume it is 1cm thick.
Circumference = 120cm = 1.2m, cross sectional area = 10-4 m2
R = ρL/A = 2.75 x 10-8 * 1.2 / 10-4 = 3.3 x 10-4 Ω
EMF
Current
Force
Power
2 0.3 0.3 0.18Vε = = × × =BLv
4/ 0.18 / 3.3 10 545Aε −= = × =i R
2 98W= =P i RAbout 0.330 C per sec(from specific heat, density)
545 0.3 2 327 NF iLB= = × × = 74 lbs!
PHY2054: Chapter 20 28
Electric GeneratorsRotate a loop of wire in a uniform magnetic field:
changing θ ⇒ changing flux ⇒ induced emfΦB = B A cos θ = B A cos(ωt)
Rotation: θ = ωt
PHY2054: Chapter 20 29
Electric GeneratorsFlux is changing in a sinusoidal manner
Leads to an alternating emf (AC generator)
cos( ) sin( )Bd d tN NBA NBA tdt dt
ω ω ωε Φ= = =
This is how electricity is generated
Water or steam (mechanical power) turns the blades of a turbine which rotates a loop
Mechanical power converted to electrical power
PHY2054: Chapter 20 30
ConcepTest: GeneratorsA generator has a coil of wire rotating in a magnetic field. If the B field stays constant and the area of the coil remains constant, but the rotation rate increases, how is the maximum output voltage of the generator affected?
(a) Increases (b) Decreases (c) Stays the same(d) Varies sinusoidally
sin( )ω ωε = NBA t
PHY2054: Chapter 20 31
Induction in Stationary CircuitSwitch closed (or opened)
Current induced in coil B (direction shown is for closing)
Steady state current in coil ANo current induced in coil B
A
B
PHY2054: Chapter 20 32
Self - InductanceConsider a single isolated coil:
Current (red) starts to flow clockwise due to the batteryBut the buildup of current leads to changing flux in loopInduced emf (green) opposes the change
d diN Ldt dt
ε Φ=− =−
L is the self-inductanceunits = “Henry (H)”
induced emf
This is a self-induced emf (also called “back” emf)
12V
PHY2054: Chapter 20 33
Inductance of SolenoidTotal flux (length l)
( )( ) 20μΦ = =BN nl B A n A li
0μ=B in
20
Bd di diN n Al Ldt dt dt
με Φ= − = − = −
20L n Alμ=
To make large inductance:Lots of windingsBig areaLong
PHY2054: Chapter 20 34
LR CircuitsInductance and resistor in series with battery of EMF V
Start with no initial current in circuitClose switch at t = 0Current is initially 0 (initial increasecauses voltage drop across inductor)
Find i(t)Resistor: ΔVR = RiInductor: ΔVL = L di/dtUse back emf of inductor
VR
L
( ) /LV V V Ldi dti tR R−Δ −
= =
PHY2054: Chapter 20 35
Analysis of LR CircuitGeneral solution
( )( )// 1 −= − tR Li V R e
Final current (maximum)
Rise from 0 withtime constant τ = L / R
PHY2054: Chapter 20 36
L-R Circuits (2)Switch off battery: Find i(t) if current starts at i0
/0
tR Li i e−= Exponential fall to 0 withtime constant τ = L / R
Initial current (maximum)
PHY2054: Chapter 20 37
Exponential Behaviorτ = L/R is the “characteristic time” of any RL circuit
Only t / τ is meaningful, just like for RC circuit
t = τCurrent falls to 1/e = 37% of maximum valueCurrent rises to 63% of maximum value
t =2τCurrent falls to 1/e2 = 13.5% of maximum valueCurrent rises to 86.5% of maximum value
t =3τCurrent falls to 1/e3 = 5% of maximum valueCurrent rises to 95% of maximum value
t =5τCurrent falls to 1/e5 = 0.7% of maximum valueCurrent rises to 99.3% of maximum value
PHY2054: Chapter 20 38
ConcepTest: Generators and MotorsA current begins to flow in a wire loop placed in a magnetic field as shown. What does the loop do?
(a) moves to the right (b) moves up (c) rotates around horizontal axis(d) rotates around vertical axis(e) moves out of the page
This is how a motor works !!
B
PHY2054: Chapter 20 39
Current is supplied from an external source of emf (battery or power supply)
Forces act to rotate the wire loop
A motor is essentially a generator operated in reverse!
Electric Motors
PHY2054: Chapter 20 40
MotorForces act to rotate the loop towards the vertical.
When loop is vertical, current switches sign and the forces reverse, in order to keep the loop in rotation.
This is why alternating current is necessary for a motor to operate.
PHY2054: Chapter 20 41
Motors Generators
Electrical ⇒ mechanical energy Mechanical ⇒ electrical energy
PHY2054: Chapter 20 42
Energy of an InductorEnergy is stored in inductor when current flows through it
Energy is stored in the B field
Similar to energy in charged capacitor
Energy is stored in the E field (between plates)
212LU Li=
221
22CqU CVC
= =
PHY2054: Chapter 20 43
Energy of SolenoidRecall B field of solenoid (current i, n turns per meter)
Inductance of solenoid
Energy of solenoid can be written in terms of B field and volume (V = lA)
0B niμ=
( )( )0 20
B nl niANL n Ali i
μμΦ
= = =
( )2 2
2 21 102 2
0 02LB BU Li n lA Al
nμ
μ μ⎛ ⎞
= = =⎜ ⎟⎝ ⎠
0B BA niAμΦ = =
Energy densityof B field
Volume
PHY2054: Chapter 20 44
Energy of Solenoid (2)Typical MRI magnet
B = 0.5 – 1.5 TR = 0.25 m, l = 2.5 m
Assume B = 1 T typically
This is a lot of energyNeeds to be held in by strong materials
2 30.5mV Al R lπ= = ≅
( )2 2
70
1 0.5 200 kJ2 2 4 10BBU Vμ π −= = ≈
× ×
PHY2054: Chapter 20 45
Gigajoule Magnet at CERNCMS experiment at CERN
p-p collisions at world’s highest energy in 2007Hope to discover new particles, find the origin of mass and new fundamental forces
Compact Muon Solenoid
PHY2054: Chapter 20 47
CMS Experiment MagnetLarge central solenoid magnet to study particle production
B = 4T, R = 3.15 m, l = 12.5 mUB = 2.6 x 109 J = 2.6 gigajoules!!
http://www.spacedaily.com/news/energy-tech-04b.html
( )( )2 2
27
0
4 3.15 12.52 2 4 10BBU Al πμ π −= = ×
× ×
PHY2054: Chapter 20 48
CMS Articles and PicturesHome page
http://cmsinfo.cern.ch/Welcome.html/
Pictures of detectorhttp://cmsinfo.cern.ch/Welcome.html/CMSdetectorInfo/CMSdetectorInfo.htmlhttp://cmsinfo.cern.ch/Welcome.html/CMSmedia/CMSmedia.html
Interesting article on solenoid, with pictureshttp://cmsinfo.cern.ch/Welcome.html/CMSdocuments/MagnetBrochure/MagnetBrochure.pdf
Other documents & pictures about CMShttp://cmsinfo.cern.ch/Welcome.html/CMSdocuments/CMSdocuments.html