The document discusses key concepts related to electromagnetic induction including: - Faraday's law states that the induced emf in a circuit is equal to the rate of change of magnetic flux through the circuit. - Lenz's law describes the direction of induced current: it will flow in a direction to oppose the change in magnetic flux that created it. - Self-induction occurs when a changing current in a circuit induces an opposing emf known as back emf. The property of self-inductance depends on geometry. - RL circuits with both resistance and inductance experience an exponential rise or decay of current over time, depending on if the switch is closed or opened, with a characteristic time constant.

1. Induced emf
• Experiment:
– When the switch is closed, the ammeter deflects
momentarily
– There is an induced emf in the secondary coil (producing
an induced current) due to the changing magnetic field
through this coil
• When there is a steady current in the primary coil, the
ammeter reads zero current
– Steady magnetic fields cannot produce a current

2. Magnetic Flux
• In general, induced emf is actually due to a change
in magnetic flux
• Magnetic flux is defined similarly
to electric flux:
Φ B = B⊥ A = BA cos θ
– Units of flux are T⋅m2 (or Wb)
– Flux can be positive or negative
• The value of the magnetic flux
through a loop/surface is
proportional to the total number
of field lines passing through the
loop/surface

3. Faraday’s Law
• Faraday’s Law says that the magnitude of the
induced emf in a circuit is equal to the time rate of
change of the magnetic flux through the circuit
• The average emf ε induced in a circuit containing N
tightly wound loops is given by: ∆Φ B
– Magnetic flux changes by amount ∆ΦB ε = −N
∆t
during the time interval ∆t
• The minus sign in the equation concerns the sense
of the polarity of the induced emf in the loop (and
how it makes current flow in the loop)
– The direction follows Lenz's law:
• The direction of the induced emf is such that it produces a current
whose magnetic field opposes the change in magnetic flux through
the loop
• The induced current tends to maintain the original flux through the
loop

4. Example Problem #20.58
The square loop shown is made
of wires with a total series
resistance of 10.0 Ω. It is placed
in a uniform 0.100-T magnetic
field directed perpendicular into
the plane of the paper. The loop,
which is hinged at each corner,
is pulled as shown until the
separation between points A and
B is 3.00 m. If this process takes
0.100 s, what is the average
current generated in the loop?
What is the direction of the
current?
Solution (details given in class):
0.121 A clockwise

5. Interactive Example Problem:
Flux and Induced emf in a Loop
Animation and solution details given in class.
(PHYSLET Physics Exploration 29.3, copyright Pearson Prentice Hall, 2004)

6. Example Problem #20.29
Find the direction of the current in the resistor R shown in the figure after each
of the following steps (taken in the order given): (a) The switch is closed. (b)
The variable resistance in series with the battery is decreased. (c) The circuit
containing resistor R is moved to the left. (d) The switch is opened.
Solution (details given in class):
(a) Right to left
(b) Right to left
(c) Left to right
(d) Left to right

7. Interactive Example Problem:
Lenz’s Law
Animation and solution details given in class.
(PHYSLET Physics Exploration 29.1, copyright Pearson Prentice Hall, 2004)

8. Applications of Faraday’s Law
• Ground fault interrupter (GFI)
• Electric guitar pickups
• Magnetoencephalography (brain),
magnetocardiograms (heart and surrounding nerves)

9. Motional emf
• Consider a straight conducting bar
moving perpendicularly to the direction l
of a uniform external magnetic field
– Free charges (e–) experience downward
force F
– Creates a charge separation in the conductor
– Charge separation leads to a downward electric field E
– Eventually equilibrium is reached and the (downward)
magnetic force qvB on the electrons balances the (upward)
electric force qE
• qE = qvB  E = vB
• ∆V = El (since E uniform)  ∆V = El = Blv
– A potential difference is maintained across the conductor as
long as there is motion through the field
• If motion is reversed, polarity of potential difference is also reversed

10. Motional emf
• Now consider the same bar as part of
a closed conducting path
– Since the area of the current loop changes
as the bar moves, an induced emf is
created due to a changing magnetic flux
through the loop
– An induced current is created
– The direction of the current is counterclockwise (Lenz’s law)
– Assume bar moves a distance ∆x in time ∆t
• ∆ΦB = B∆A = Bl∆x
• From Faraday’s Law (and noting that N = 1):
∆Φ B ∆x
ε = = Bl = Blv
∆t ∆t
(motional emf)

11. Motional emf
• Notes on this experiment:
– The moving bar acts like a battery with an emf given by the
previous formula
– Although the current tends to deplete the
accumulated charge at each end of the
bar, the magnetic force pumps more
charge to maintain a constant potential
difference
– At right is the equivalent circuit diagram
– The work done by the force pulling the rod is the source of
the electrical energy
– This setup is not practical; a more practical use of motional
emf is through a rotating coil of wire (electric generators)
– Note that a battery can be used to drive a current, which
establishes a magnetic field, which can propel the bar

12. Inductive Magnetoreception in Sharks
Physics Today, March 2008

13. Launching Mechanism (“Rail Gun”)

14. Example Problem #20.21
A car has a vertical radio antenna 1.20 m long. The
car travels at 65.0 km/h on a horizontal road where the
Earth’s magnetic field is 50.0 µT, directed toward the
north and downwards at an angle of 65.0° below the
horizontal. (a) Specify the direction the car should
move in order to generate the maximum motional emf
in the antenna, with the top of the antenna positive
relative to the bottom. (b) Calculate the magnitude of
this induced emf.
Solution (details given in class):
(a) Toward the east
(b) 4.58 × 10–4 V

15. Electric Generators
• Alternating current (AC) generators f = 60 Hz (U.S.)
ε = NBAω sin ωt (ω = 2πf )
N = # turns in loop coil
B = magnetic field strength
A = Area of the loop coil
ω = angular rotation speed
εmax = NBAω (plane of loop
parallel to magnetic field)
(θ = ωt)
εmin = 0 (plane of loop
perpendicular to magnetic field)
• Direct current (DC) generators
(note similarity to DC
motor – Chap. 19)

16. Motors and Back emf
• Generators convert mechanical energy to electrical
energy
• Motors convert electrical
energy to mechanical energy
 generators run in reverse
• As the coil in the motor
rotates, the changing
magnetic flux through it
induces an emf
– This emf acts to reduce the current in the coil (in
agreement with Lenz’s law)
– For this reason, the emf is called a back emf
– Power requirements for starting a motor or running it under
heavy loads larger than running under average loads

17. Self-Inductance
• Consider the effect of current flow in
a simple circuit with a battery and a
resistor
– Faraday’s law prevents the current from
reaching its maximum value (ε / R)
immediately after switch is closed
• Magnetic flux through the loop changes as current increases
• Resulting induced emf acts to reduce total emf of circuit
– When the rate of current change lessens as current
increases, induced emf decreases
• Another example:
increasing and
decreasing current in
a solenoid coil

18. Self-Inductance
• These effects are called self-induction
• The emf that is produced is called self-induced emf
(or back emf) ∆I
ε = −L
∆t
– L is a constant called the inductance
– The minus sign indicates that a changing current induces
an emf that is in opposition to that change
– Units of inductance is the henry (H)  1 H = 1 V⋅s / A
• An expression for inductance can be found by
equating Faraday’s law with the above:
∆Φ B ΦB
L=N =N
∆I I

19. RL Circuits
• A circuit element having a large inductance is called
an inductor
∆I
• From ε L = − L ∆t we see that an inductor acts as a
current stabilizer
– The larger the inductance in a circuit, the larger the
opposition to the rate of change of the current
– Remember that resistance is a measure of the opposition
to current
• Consider a series circuit
consisting of a battery,
resistor, and inductor
– Current changes most rapidly
when switch is first closed
• Back emf largest at this point (Series RL circuit)

20. RL Circuits
• Using calculus, we can derive an
expression for the current in the
circuit as a function of time
ε
(rise of I=
R
(1 − e
− Rt / L
)
current)
– Similar to charge q as a function of
time for a charging capacitor
– Like RC circuits, there is a time constant for the circuit:
L
τ=
R
• When the switch is opened, the current in the circuit
decreases exponentially with time ε − Rt / L
I= e
(decay of current R
)

21. Example Problem #20.53
An 820-turn wire coil of resistance 24.0 Ω is
placed on top of a 12500-turn, 7.00-cm-long
solenoid as shown. Both coil and solenoid have
cross-sectional areas of 1.00×10–4 m2. (a) How
long does it take the solenoid current to reach
0.632 times its maximum value? (b) Determine
the average back emf caused by the self-
inductance of the solenoid during this interval.
(c) The magnetic field produced by the solenoid
at the location of the coil is ½ as strong as the
field at the center of the solenoid. Determine the
average rate of change in magnetic flux through
each turn of the coil during the stated interval.
(d) Find the magnitude of the average Solution (details given in class):
induced current in the coil. (a) 20.0 ms
(b) 37.9 V
(c) 1.52 mV
(d) 51.8 mA