1. General Physics (PHY 2140) Lecture 9 Electricity and Magnetism
Induced voltages and induction
Magnetic flux and induced emf
Faraday’s law
Motional EMF
Lenz’s Law
Applications
Self-Inductance
RL Circuits (maybe)
Chapter 20
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2. Lightning Review Last lecture: Magnetism
Application of magnetic forces
Ampere’s law
Current loops and solenoids
Induced voltages and induction
Magnetic flux
Review Problem: A rectangular loop is placed in a uniform magnetic field with the plane of the loop parallel to the direction of the field. If a current is made to flow through the loop in the sense shown by the arrows, the field exerts on the loop:
1. a net force.
2. a net torque.
3. a net force and a net torque.
4. neither a net force nor a net torque.
Recall:
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3. Magnetic Field of a current loop
Magnetic field produced by a wire can be enhanced by having the wire in a loop.
Dx1 I B
Dx2
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4. Solenoid Magnet
Field lines inside a solenoid magnet are parallel, uniformly spaced and close together.
The field inside is uniform and strong.
The field outside is non uniform and much weaker.
One end of the solenoid acts as a north pole, the other as a south pole.
For a long and tightly looped solenoid, the field inside has a value:
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5. Solenoid Magnet n = N/l : number of (loop) turns per unit length.
I : current in the solenoid.
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6. Chapter 20: Introduction
Previous chapter: electric currents produce magnetic fields (Oersted’s experiments)
Is the opposite true: can magnetic fields create electric currents?
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7. 20.1 Induced EMF and magnetic flux
Definition of Magnetic Flux
Just like in the case of electric flux, consider a situation where the magnetic field is uniform in magnitude and direction. Place a loop in the B-field.
The flux, F, is defined as the product of the field magnitude by the area crossed by the field lines.
where is the component of B perpendicular to the loop, q is the angle between B and the normal to the loop.
Units: T·m2 or Webers (Wb)
The value of magnetic flux is proportional to the total number of magnetic field lines passing through the loop.
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8. Problem: determining a flux
A square loop 2.00m on a side is placed in a magnetic field of strength 0.300T. If the field makes an angle of 50.0° with the normal to the plane of the loop, determine the magnetic flux through the loop.
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9. From what we are given, we use
A square loop 2.00m on a side is placed in a magnetic field of strength 0.300T. If the field makes an angle of 50.0° with the normal to the plane of the loop, determine the magnetic flux through the loop.
Solution:
Given:
L = 2.00 m
B = T
q = 50.0˚
Find:
F=?
From what we are given, we use
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10. 20.1 Induced EMF and magnetic flux
Faraday’s experiment
Two circuits are not connected: no current?
However, closing the switch we see that the compass’ needle moves and then goes back to its previous position
Nothing happens when the current in the primary coil is steady
But same thing happens when the switch is opened, except for the needle going in the opposite direction…
Picture © Molecular Expressions
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What is going on?

11. 20.2 Faraday’s law of induction
Induced current I N S v
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12. 20.2 Faraday’s law of inductionv B I N S I
A current is set up in the circuit as long as there is relative motion between the magnet and the loop.
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13. Does there have to be motion?
AC Delco - +
1 volt I I
(induced)
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14. Does there have to be motion?
AC Delco - +
1 volt
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15. Does there have to be motion?
AC Delco - +
1 volt I
(induced)
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16. Does there have to be motion?
NO!!
AC Delco - +
1 volt
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17. Maybe the B-field needs to change…..v B
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18. Maybe the B-field needs to change…..v B
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19. Maybe the B-field needs to change…..v B I
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20. Faraday’s law of magnetic induction
In all of those experiment induced EMF is caused by a change in the number of field lines through a loop. In other words,
The instantaneous EMF induced in a circuit equals the rate of change of magnetic flux through the circuit.
Lenz’s Law: The polarity of the induced emf is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop. That is, the induced current tends to maintain the original flux through the circuit.
Lenz’s law
The number of loops matters
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21. Applications: Ground fault interrupter Electric guitar SIDS monitor
Metal detector …
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22. Example : EMF in a loop
A wire loop of radius 0.30m lies so that an external magnetic field of strength +0.30T is perpendicular to the loop. The field changes to -0.20T in 1.5s. (The plus and minus signs here refer to opposite directions through the loop.) Find the magnitude of the average induced emf in the loop during this time.
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23. and after the field changes the flux is
A wire loop of radius 0.30m lies so that an external magnetic field of strength +0.30T is perpendicular to the loop. The field changes to -0.20T in 1.5s. (The plus and minus signs here refer to opposite directions through the loop.) Find the magnitude of the average induced emf in the loop during this time.
Given:
r = 0.30 m
Bi = 0.30 T
Bf = T
Dt = 1.5 s
Find:
EMF=?
The loop is always perpendicular to the field, so the normal to the loop is parallel to the field, thus cosq = 1. The flux is then
Initially the flux is
and after the field changes the flux is
The magnitude of the average induced emf is:
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24. Example 2: EMF of a flexible loop
The flexible loop in figure below has a radius of 12cm and is in a magnetic field of strength 0.15T. The loop is grasped at points A and B and stretched until it closes. If it takes 0.20s to close the loop, find the magnitude of the average induced emf in it during this time. A
X X X X
ΔΦ = BA = 0.15T × π (0.12 m)2 = T∙m2
Δt = 0.20 s Then, emf = ΔΦ/ Δt = V B
emf = V
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25. 20.3 Motional EMF l v B F
Let's consider a conducting bar moving perpendicular to a uniform magnetic field with constant velocity v.
This force will act on free charges in the conductor. It will tend to move negative charge to one end, and leave the other end of the bar with a net positive charge.
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26. Motional EMF
The separated charges will create an electric field which will tend to pull the charges back together.
When equilibrium exists, the magnetic force, F=qvB, will balance the electric force, F=qE, such that a free charge in the bar will feel no net force.
Thus, at equilibrium, E = vB. The potential difference across the ends of the bar is given by DV=El or
A potential difference is maintained across the conductor as long as there is motion through the field. If the motion is reversed, the polarity of the potential difference is also reversed.
(recall our velocity selector example)
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27. Motional EMF – conducting railsv Dx B
We can apply Faraday's law to the complete loop. The change of flux through the loop is proportional to the change of area from the motion of the bar:
or (Faraday’s law)
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current
Motional EMF

28. Example: wire in the magnetic field
Over a region where the vertical component of the Earth's magnetic field is 40.0µT directed downward, a 5.00 m length of wire is held in an east-west direction and moved horizontally to the north with a speed of 10.0 m/s. Calculate the potential difference between the ends of the wire, and determine which end is positive.
EMF = Blv = 40.0 µT × 5.00 m × 10.0 m/s = V
Use right hand rule and F = qv × B to determine the polarity.
Answer: west end is positive.
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29. 20.4 Lenz’s law revisited
Application of Lenz's law will tell us the direction of induced currents, the direction of applied or produced forces, and the polarity of induced emf's.
Lenz's law says that the induced current will produce magnetic flux opposing this change. To oppose an increase into the page, it generates magnetic field which points out of the page, at least in the interior of the loop. Such a magnetic field is produced by a counterclockwise current (use the right hand rule to verify).
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30. Lenz’s law: energy conservation
We arrive at the same conclusion from energy conservation point of view
The preceding analysis found that the current is moving ccw. Suppose that this is not so.
If the current I is cw, the direction of the magnetic force, BlI, on the sliding bar would be right.
This would accelerate the bar to the right, increasing the area of the loop even more.
This would produce even greater force and so on.
In effect, this would generate energy out of nothing violating the law of conservation of energy.
Our original assertion that the current is cw is not right, so the current is ccw!
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31. S S S N S N N v v The induced flux seeks to counteract the change.
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32. Example: direction of the current
Find the direction of the current induced in the resistor at the instant the switch is closed.
Induced current B
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33. Applications of Magnetic Induction
Tape / Hard Drive / ZIP Readout
Tiny coil responds to change in flux as the magnetic domains (encoding 0’s or 1’s) go by.
Question: How can your VCR display an image while paused?
Credit Card Reader
Must swipe card
 generates changing flux
Faster swipe  bigger signal
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34. 20.5 Generators
Generators and motors are two of the most important applications of induced emf (magnetic inductance).
A generator is something that converts mechanical energy to electrical energy.
Alternating Current (AC) generator
Direct Current (DC) generator
A motor does the opposite, it converts electrical energy to mechanical energy.
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35. AC generator Compute EMF It is only generated in BC and DA wires
EMF generated in BC and DA would be
Thus, total EMF is
If the loop is rotating with w a A B
v sin q v θ B A
as v=rw=aw/2 and q = wt
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36. AC generator (cont) Generalize the result to N loops
where we also noticed that A=la
Note: is reached when wt=90˚ or 270˚ (loop parallel to the magnetic field)
EMF generated by the AC generator
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37. DC generator
By a clever change to the rings and brushes of the ac generator, we can create a dc generator, that is, a generator where the polarity of the emf is always positive. The basic idea is to use a single split ring instead of two complete rings. The split ring is arranged so that, just as the emf is about to change sign from positive to negative, the brushes cross the gap, and the polarity of the contacts is switched. The polarity of the contacts changes in phase with the polarity of the emf -- the two changes essentially cancel each other out, and the emf remains always positive. The emf still varies sinusoidally during each half cycle, but every half cycle is a positive emf.
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38. DC Generator
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39. Motors
A motor is basically a generator running in reverse. A current is passed through the coil, producing a torque and causing the coil to rotate in the magnetic field. Once turning, the coil of the motor generates a back emf, just as does the coil of a generator. The back emf cancels some of the applied emf, and limits the current through the coil.
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40. Example: coil in magnetic field
A coil of area 0.10 m² is rotating at 60 rev/s with its axis of rotation perpendicular to a 0.20T magnetic field. (a) If there are 1000 turns on the coil, what is the maximum voltage induced in the coil? (b) When the maximum induced voltage occurs, what is the orientation of the coil with respect to the magnetic field?
Recall:   2 f
1000(0.20)(0.10)(260) = 7540 V
Max when loop parallel to the magnetic field.
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41. Eddy currents (application)
Magnetic Levitation (Maglev) Trains
Induced surface (“eddy”) currents produce field in opposite direction
 Repels magnet
 Levitates train
Maglev trains today can travel up to 310 mph
 Twice the speed of Amtrak’s fastest conventional train!
May eventually use superconducting loops to produce B-field
 No power dissipation in resistance of wires! N S
rails
“eddy” current
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42. 20.6 Self-inductance
When a current flows through a loop, the magnetic field created by that current has a magnetic flux through the area of the loop.
If the current changes, the magnetic field changes, and so the flux changes giving rise to an induced emf. This phenomenon is called self-induction because it is the loop's own current, and not an external one, that gives rise to the induced emf.
Faraday’s law states
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43. where L is called the inductance of the device Units: SI: henry (H)
The magnetic flux is proportional to the magnetic field, which is proportional to the current in the circuit
Thus, the self-induced EMF must be proportional to the time rate of change of the current
where L is called the inductance of the device
Units: SI: henry (H)
If flux is initially zero,
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44. Example: solenoid
A solenoid of radius 2.5cm has 400 turns and a length of 20 cm. Find (a) its inductance and (b) the rate at which current must change through it to produce an emf of 75mV.
= (4p x 10-7)(160000)(2.0 x 10-3)/(0.2) = 2 mH
= -(75 x 10-3)/ (2.0 x 10-3) = 37.5 A/s
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45. Inductor in a Circuit
Inductance can be interpreted as a measure of opposition to the rate of change in the current
Remember resistance R is a measure of opposition to the current
As a circuit is completed, the current begins to increase, but the inductor produces an emf that opposes the increasing current
Therefore, the current doesn’t change from 0 to its maximum instantaneously
Maximum current:
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46. 20.7 RL Circuits Recall Ohm’s Law to find the voltage drop on R
We have something similar with inductors
Similar to the case of the capacitor, we get an equation for the current as a function of time (series circuit).
(voltage across a resistor)
(voltage across an inductor)
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47. RL Circuit (continued)
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48. 20.8 Energy stored in a magnetic field
The battery in any circuit that contains a coil has to do work to produce a current
Similar to the capacitor, any coil (or inductor) would store potential energy
Summary of the properties of circuit elements.
Resistor
Capacitor
Inductor
units
ohm, W = V / A
farad, F = C / V
henry, H = V s / A
symbol R C L
relation
V = I R
Q = C V
emf = -L (DI / Dt)
power dissipated
P = I V = I² R = V² / R
energy stored
PEC = C V² / 2
PEL = L I² / 2
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49. Example: stored energy
A 24V battery is connected in series with a resistor and an inductor, where R = 8.0W and L = 4.0H. Find the energy stored in the inductor (a) when the current reaches its maximum value and (b) one time constant after the switch is closed.
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