Short Version : 27. Electromagnetic Induction. 27.1. Induced Currents. 4 results from Faraday / Henry (1831). v = 0, I = 0. v > 0, I > 0. 1. Current induced in coil by moving magnet bar. v >> 0, I >> 0. v < 0, I < 0.
4 results from Faraday / Henry (1831)
v = 0, I = 0
v > 0, I > 0
1. Current induced in coil by moving magnet bar.
v >> 0, I >> 0
v < 0, I < 0
2. Moving the coil instead of the magnet gives the same result.
3. An induced current also results when a current-carrying circuit replaces the magnet.
4. A current is also induced when the current in an adjacent circuit changes.
changing B induces currents (electromagnetic induction)
For a uniform B on a flat surface:
Move magnet right
more lines thru loop
A solenoid of circular cross section has radius R,
consists of n turns per unit length, and carries current I.
Find the magnetic flux through each turn of the solenoid.
B out of plane
A long, straight wire carries current I.
A rectangular wire loop of dimensions l by w lies in a plane containing the wire, with its closest edge a distance a from the wire, and its dimension l parallel to the wire.
Find the magnetic flux through the loop.
Area element for integration
Faraday’s law of induction:
The induced emf in a circuit is proportional to the rate of change of magnetic flux through any surface bounded by that circuit.
C is CCW about S.
Note: dB/dt can be due to
A wire loop of radius 10 cm has resistance 2.0 .
The plane of the loop is perpendicular to a uniform B that’s increasing at 0.10 T/s.
Find the magnitude of the induced current in the loop.
Two parallel conducting rails a distance l apart are connected at one end by a resistance R.
A conducting bar completes the circuit, joining the two rails electrically but free to slide along.
The whole circuit is perpendicular to a uniform B, as show in figure.
Find the current when the bar is pulled to the right with constant speed v.
Let x = 0 be at the left end of rail.
Direction of emf is to oppose magnet’s motion.
RH rule: thumb // m.
Loop ~ magnet with N to left.
Magnet moving right
Lenz’s law :
Direction of induced emf is such that B created by the induced current opposes the changes in that created the current.
RH rule: thumb // m.
Loop ~ magnet with S to left.
Magnet moving left
Motional emf: induced emf due to motion of conductor in B.
Square loop of sides L & resistance R pulled with constant speed v out of uniform B.
Force on e:
Force on current carrying wire:
Work done is used to heat up circuit ( E conservation ).
World electricity generation ~ 2TW.
Rotating loop changes & induces emf.
Rotating slip rings.
Sinusoidal AC output
Work required due to Lenz’s law.
Rotating conducting loop
Hand-cranked generator ~ 100W
An electric generator consists of a 100-turn circular coil 50 cm in diameter.
It’s rotated at f = 60 rev/s to produce standard 60 Hz alternating current.
Find B needed for a peak output voltage of 170 V,
which is the actual peak in standard 120 V household wiring.
EM induction is basis of magnetic recording ( audio, video, computer disks, …).
Modern hard disks:
Information stored in magnetization pattern
Swiping a credit card.
Patterns of magnetization on the strip induce currents in the coil.
Eddy current: current in solid conductor induced by changing .
Usage: non-frictional brakes for rotating saw blades, train wheels, …
Application: Metal Detectors
Nothing between coils
Weak I : alarm.
Metal between coils
B of induced I points out of page
Setting n // Bin
C is CCW & < 0
Bin d /d t < 0
E > 0
I is CCW
RH rule gives CCW I
E > 0
Changing current in one circuit induces an emf in the other.
two coils are wound on same iron core.
Transformers, ignition coil, battery chargers, …
Changing current induces emf in own circuit & opposes further changes.
Inductors frequency generator / detector …
[ L ] = T m2 / A = Henry
A long solenoid of cross section area A and length l has n turns per unit length.
Find its self-inductance.
B of solenoid:
Rapid switching of inductive devices can destroy delicate electronic devices.
dI /d t < 0
+ (higher voltage)
+ (higher voltage)
A 5.0-A current is flowing in a 2.0-H inductor.
The current is then reduced steadily to zero over 1.0 ms.
Find the magnitude & direction of the inductor emf during this time.
+ here ( E > 0 ) helps keep I flowing
Current through inductor can’t change instantaneously.
Switch just closed :
I = 0, dI/dt 0; | EL | = E0
L ~ open circuit.
Long after switch closing:
I 0, dI/dt = 0; EL= 0;
L ~ wire.
Switch open: I = 0
I V = IR
But rate is
EL< 0 ; | EL |
Inductive time constant = L / R
c.f. capacitive time constant = RC
A large electromagnet used for lifting scrap iron has self-inductance L = 56 H.
It’s connected to a constant 440-V power source;
the total resistance of the circuit is 2.8 .
Find the time it takes for the current to reach 75% of its final value.
Switch at A, I.
Short times: IL can’t change instantaneously.
Long times: EL = 0 ; inductor wire.
Verify that the current in just after the switch is reopened has the value indicated
Immediately after the switch is closed:
L ~ open circuit.
Long time after the switch is closed:
Immediately after 2nd switch opening :
L ~ short circuit.
I in L ~ continues.
Any B contains energy.
This eruption of a huge prominence from the sun’s surface releases energy stored in magnetic fields.
Power from battery
Power taken by inductor
Energy stored in inductor:
In practice, Cu / Ag are incorporated into the superconducting wires to reduce R.
Solenoid with length l & cross-section area A :
Magnetic Energy Density :
c.f. electric energy density :
EMF acts to separate charges:
Battery: chemical reaction
Motional emf: F = v B.
Stationary loop in changing B : induced E
Induced E forms loop.
Static E begins / ends on charge.
A long solenoid has circular cross section of radius R.
The solenoid current is increasing, & as a result so is B in solenoid.
The field strength is given by B = b t, where b is a constant.
Find the induced E outside the solenoid, a distance r from the axis.
Symmetry E lines are circles.
Loop for Faraday’s law
S out C ccw
For stationary charges (electrostatics) :
W against E
E is conservative
Induced fields (electromagnetics) :
E does W
E is non-conservative
The figure shows three resistors in series surrounding an infinitely long solenoid with a changing magnetic field;
the resulting induced electric field drives a current counterclockwise, as shown.
Two identical voltmeters are shown connected to the same points A and B.
What does each read?
Explain any apparent contradiction.
Hint: this is a challenging question!
VA VB = IR
VA VB = 2IR
Classical model of diamagnetism (not quite right)
Superconductor is a perfect diamagnet (Meissner effect).
B = 0: net = 0
This e slows down.
This e speeds up.