Lecture 26
EMF. Induced field.
Displacement current.
ACT: Two rings

DEMO:
Jumping rings

Two copper rings with the same geometry move toward identical
magnets with the same v...
In-class example: AC generator
A coil of copper wire consists of 120 loops of wire wound around a 10
cm × 20 cm rectangula...
This is an AC generator!

ΦB = BA cos ( ωt )

ε = NBAω sin ( ωt )
AC generator
Water turns wheel
 rotates magnet
 changes flux
 induces emf
 drives current

DEMOs:
Loop rotating betwee...
Eddy currents
Pull a sheet of metal through B-field

F
Induced emf due to change in flux
→ currents still generated but
pa...
Applications: Damping
Eddy currents induced in a copper plate when it passes near
magnets.
Low resistivity ↔ large current...
Metal detectors
• Pulse current through primary coil
– B-field which changes with time
– induces currents in a piece of ne...
Back to the EMF
EMF of a battery (from Phys 221):

r r
ε = V+ −V− = − ∫ E × =
dl
+

−

∫

−

+

r r
E ×
dl

Integrating in...
Motional emf
A loop moves towards a magnet
⇒ a current is induced.

S

N

v

r
r r
Cause: Magnetic force on the moving cha...
ACT: E-field in an open circuit
What is the direction of the E-field in the moving
conductor?
A. Left
B. Right
C. Up
D. Do...
Induced electric field
A magnet moves towards a loop ⇒ a current is induced.
S

N

v

No magnetic forces involved. ⇒ There...
Faraday’s law links E andB fields
q

generates E-field

q
experiences force

changing B-field
generates an E-field

moving...
Two types of E fields
1. E field produced by charges
• Lines begin/end on charges

r r
•
Ñ ×dl = 0 (on a closed loop)
∫E

...
Escher depiction of nonconservative emf
Charging capacitor
When the switch in the circuit below is closed, the capacitor
begins charging. While it is charging, is...
Close up of the capacitor region:

I
Current intercepting
the green area

Ampere’s law for the loop shown:

r r
Ñ ×dl = µ0...
Displacement current
There is not current, but there is a time-changing
electric field between the plates:

E =

q
ε 0A

⇒...
E as a source of B fields
Time-dependent E fields are a source of B fields!
r r

d ΦE
B × = µ0  I + ε 0
dl
Ñ
∫

dt


...
The complete picture
q

generates E-field

changing B-field
generates an E-field

moving q

q
experiences force

changing ...
Maxwell’s equations
r
r q
Ñ ×dA = ε
∫E
0

Gauss’s law for E

r
r
Ñ ×dA = 0
∫B

Gauss’s law for B

r r
d ΦB
Ñ ×dl = − dt
∫E...
In the absence of sources
The symmetry is then very impressive:

r
r
Ñ ×dA = 0
∫E

r
r
Ñ ×dA = 0
∫B

r r
d ΦB
Ñ ×dl = − dt...
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Lecture 26 emf. induced fields. displacement currents.

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Lecture 26 emf. induced fields. displacement currents.

  1. 1. Lecture 26 EMF. Induced field. Displacement current.
  2. 2. ACT: Two rings DEMO: Jumping rings Two copper rings with the same geometry move toward identical magnets with the same velocity as shown. The ring in case 2, though, has a small slit. Compare the magnitudes of: • the induced emf’s in the rings: A. ε1 < ε2 B. ε1 = ε2 C. ε1 > ε2 EMF is independent of the actual ring. It would be the same for a wooden ring, or even in vacuum! v N S 1 v N S 2 • the magnetic force on the rings: A. F1 < F2 B. F1 = F2 C. F1 > F2 No current in case 2 because ring is open. F2 = 0
  3. 3. In-class example: AC generator A coil of copper wire consists of 120 loops of wire wound around a 10 cm × 20 cm rectangular form. If this coil is used to generate an induced emf by rotating it in a 0.3 T field at 60 Hz, what is the maximum emf that can be produced? r r A B r r A. 33 V ωt ΦB = NB ×A B. 43 V = NBA cos ( ωt ) C. 120 V D. 217 V E. 27 100 V d ΦB ε =− = ωNBA sin ( ωt ) dt ε max = ωNBA  turns 2π rad  2 =  60 ÷( 120 ) ( 0.3 T ) 0.1 × 0.2 m s 1 turn   = 271 V ( )
  4. 4. This is an AC generator! ΦB = BA cos ( ωt ) ε = NBAω sin ( ωt )
  5. 5. AC generator Water turns wheel  rotates magnet  changes flux  induces emf  drives current DEMOs: Loop rotating between electromagnets Hand-driven generator with bulbs
  6. 6. Eddy currents Pull a sheet of metal through B-field F Induced emf due to change in flux → currents still generated but path they take not well-defined → eddy currents External force required to pull sheet because induced current opposes change. Current through bulk → heating → energy loss F
  7. 7. Applications: Damping Eddy currents induced in a copper plate when it passes near magnets. Low resistivity ↔ large currents → Magnets exert force against motion → Plate is slowed down magnet v copper magnet This effect does not “wear down” (like rubbing surfaces do). It is used as a brake system in roller coasters and alike. To reduce eddy currents when undesirable, prevent currents from flowing (cutting slots or laminating material). DEMOs: Eddy currents: pendulum and magnets through tube
  8. 8. Metal detectors • Pulse current through primary coil – B-field which changes with time – induces currents in a piece of nearby metal • Induced currents generate B-field which changes in time – induces currents in coils of metal detector – sets off signal, alarm…
  9. 9. Back to the EMF EMF of a battery (from Phys 221): r r ε = V+ −V− = − ∫ E × = dl + − ∫ − + r r E × dl Integrating in the direction of the current r r ε = ∫E × dl I + -
  10. 10. Motional emf A loop moves towards a magnet ⇒ a current is induced. S N v r r r Cause: Magnetic force on the moving charges in the loop: F = qv × B r r r r r dl dl Work: W = ∫ F × = ∫ q v × B × ( ) If this was an electric force, the corresponding emf would be W ε= = q r r ∫ F ×dl q Motional emf = ∫( r r r v ×B × dl ε =∫ ) ( r r r v ×B × dl ) Note: This does not come from an electric field.
  11. 11. ACT: E-field in an open circuit What is the direction of the E-field in the moving conductor? A. Left B. Right C. Up D. Down B constant with time v Force points left. Positive charge accumulates at left end. Electric field points right. But this a regular electrostatic E field produced by charges.
  12. 12. Induced electric field A magnet moves towards a loop ⇒ a current is induced. S N v No magnetic forces involved. ⇒ There must be an (induced) electric r field! E r E I r E r E r r r r d ε = ∫E × = − dl ∫ B ×dA dt Changing B-field induces an E-field. No charges involved!!
  13. 13. Faraday’s law links E andB fields q generates E-field q experiences force changing B-field generates an E-field moving q generates B-field moving q experiences force
  14. 14. Two types of E fields 1. E field produced by charges • Lines begin/end on charges r r • Ñ ×dl = 0 (on a closed loop) ∫E ⇒ Conservative electric field 2. E field produced by B field that changes with time • lines are loops that do not begin/end r r • Ñ ×dl = ε ( ≠ 0 ) ∫E ⇒ Non conservative electric field B decreasing with time curlyE-field induced Both types of E-fields exert forces on charges
  15. 15. Escher depiction of nonconservative emf
  16. 16. Charging capacitor When the switch in the circuit below is closed, the capacitor begins charging. While it is charging, is there a current between the plates? No. I I
  17. 17. Close up of the capacitor region: I Current intercepting the green area Ampere’s law for the loop shown: r r Ñ ×dl = µ0I ∫B But what is I decide to use this “bag” as the surface delimited by the loop? What is the current intercepting it?? I
  18. 18. Displacement current There is not current, but there is a time-changing electric field between the plates: E = q ε 0A ⇒ q = ε 0AE = ε 0 ΦE Maxwell proposed to complete Ampere’s law with an d ΦE additional “current”: dq ID = I dt = ε0 dt d ΦE ID = ε 0 dt r r Ñ ×dl = µ0 ( I + ID ) ∫B In this case, we have ID = I between the plates.
  19. 19. E as a source of B fields Time-dependent E fields are a source of B fields! r r  d ΦE B × = µ0  I + ε 0 dl Ñ ∫  dt   ÷ ÷ 
  20. 20. The complete picture q generates E-field changing B-field generates an E-field moving q q experiences force changing E-field generates a B-field generates B-field moving q experiences force
  21. 21. Maxwell’s equations r r q Ñ ×dA = ε ∫E 0 Gauss’s law for E r r Ñ ×dA = 0 ∫B Gauss’s law for B r r d ΦB Ñ ×dl = − dt ∫E r r  d ΦE Ñ ×dl = µ0  I + ε 0 dt ∫B   Faraday’s law  ÷ ÷  Ampere’s law
  22. 22. In the absence of sources The symmetry is then very impressive: r r Ñ ×dA = 0 ∫E r r Ñ ×dA = 0 ∫B r r d ΦB Ñ ×dl = − dt ∫E r r d ΦE B × = µ0 ε 0 dl Ñ ∫ dt This is 1/c2 (speed of light in vacuum)!!! There has to be some relation to light here…

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