November 03, 2014
ELECTROMAGNETISM
Electromagnetic Induction 1
November 03, 2014
Electromagnetic Induction and Faraday’s Law
R. W. Es. of Induction
Induced EMF in a Moving Conductor
Magnetic Flux
C. D. of MOTIONAL EMF Magnetic Flux Problems
Motional EMF Problems
Induced EMF
Faraday’s Law of Induction
Changing Magnetic Flux Produces an Electric Field
Electric Generators
Lenz’s Law
C. D. Lenz's Law
C. D. of Faraday's Law
Transformers and Transmission of Power
Applications of Induction
Faraday's Law Problems
Summary of Electromagnetic Induction
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REAL WORLD EXAMPLES for ELECTROMAGNETISM
COMMON EXPERIENCES:
NATURE
BIOLOGY:
Human Body
TECHNOLOGY:
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MAGNETIC FLUX DENSITY
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Where:
Φ is the magnetic flux, measured in wb 2 A is the area cut by B, measured m
B is the magnetic field, measured in wb/m2 1T=1wb/m2
Θ
is the smallest angle between the vector Normal to A and B when the tails coincide Φ is max when Θ is zero, then B is entirely perpendicular to A Φ is min when Θ is 90 or 180, then B is entirely parallel to A
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In the below diagram, the magnetic field ( blue) is perpendicular to the plane of the loop of wire ( orange) and parallel to its normal (red) so the flux is just given by ΦB= BA. If the flux passes directly though the loop of wire, perpendicular to it, then it is a maximum and ΦB= BA.
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In this case, the magnetic field is parallel to the plane of the loop of wire and perpendicular to its normal, so the flux is just given by ΦB = 0. If the magnetic field lines don't go through the loop of wire, there is no flux.
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Faraday’s Law of Induction
The magnetic flux is is proportional to the total number of lines passing through the loop.
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FARADAY'S LAW of INDUCTION:
INDUCED EMF or CURRENT 19
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Induced EMF
200 years ago, Faraday looked for evidence that a magnetic field would induce an electric current with this apparatus:
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MAGNETIC FLUX INDUCED EMF/CURRENT
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Magnetic flux will change if the area of the loop changes:
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Induced EMF
He found no evidence when the current was steady, but did see a current induced when the switch was turned on or off.
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Faraday’s Law of Induction
The induced emf in a wire loop is proportional to the rate of change of magnetic flux through the loop.
Magnetic flux is defined by: ΦB = B⊥ A
Unit of magnetic flux: weber, Wb.
1 Wb = 1 T m 2
Φ is the Greek letter "phi" and stands for "flux", or flow.
Φ B represents magnetic flux, or the flow of magnetic field through a surface.
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Concept Development of Faraday's Law
CP 1 The graph gives the magnitude B (t) of a uniform magnetic field that exists throughout a conducting loop, perpendicular to the plane of the loop. Rank the five regions of the graph according to the magnitude of the emf induced in the loop, greatest first.
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b, d+e, a=c=0 b, d+e, a=c=0 27
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Magnetic Flux Problems
1
What is the flux through a loop of wire if a magnetic field of 0.40 T is perpendicular to it and its area is 5.0 m2? Φ B = B⊥A
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2
What is the flux through a loop of wire if a magnetic field of 0.30 T is perpendicular to it and its radius is 2.0 m? Φ B = B⊥A
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3
What is the magnitude of the magnetic flux through the loop of wire shown below. The magnetic field is 1.0 T and the area of the loop of wire is 5.0 m 2.
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4
What is the magnitude of the magnetic flux through the loop of wire shown. The magnetic field is 1.0 T and the area of the loop of wire is 5.0 m 2.
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Faraday’s Law of Induction
Faraday’s law of induction:
E is the induced emf (electromovtive force) and is measured in volts (V)
N is the number of loops of wire in the coil
∆ΦB represents the change in the magnetic flux and is measured in Webers (Wb)
∆t is the time interval during which the flux changed and is measured in seconds (s)
the "­" sign has to do with the direction of the emf, and will be discussed later
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Faraday’s Law of Induction
The minus sign tells us the direction of the induced EMF.
Let's get back to that a little later, for right now let's determine the magnitude of the induced EMF.
For instance, let's figure out the induced emf in a 8.0 m2 coil if it is in a perpendicular 0.40T magnetic field that disappears over a time interval of 2.0s? (Let's assume there's just a single loop of wire.)
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What is the induced emf in a 8.0 m 2 coil (consisting of one loop) if it is perpendicular to a 0.40T magnetic field that disappears over a time interval of 2.0s?
N = 1
A = 8.0 m2
B0 = 0.4T ΦBo = B0A = (0.4T)(8.0 m2) = 3.2 Weber
Bf = 0 ΦBf = B0A = (0)(8.0 m2) = 0
∆t = 2.0s
Φ
E = ­N ∆ B
∆t
E = ­N
(ΦBf ­ ΦBo )
∆t
(0 ­ 3.2Wb)
E = ­1 = 1.6V
2.0s
Alternatively...
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What is the induced emf in a 8.0 m 2 coil (consisting of one loop) if it is perpendicular to a 0.40T magnetic field that disappears over a time interval of 2.0s?
N = 1
A = 8.0 m2
B0 = 0.4T
Bf = 0
∆t = 2.0s
Φ
E = ­N ∆ B
∆t
E = ­N ∆BA
∆t
E = ­NA ∆B
∆t
E = ­NA Bf­B0
∆t
(0 ­ 0.40T)
E = ­1(8.0 m 2) = 1.6V
2.0s
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The second approach works because anything that is not changing can be put in front of the delta symbol. For instance, if A is constant while B is changing:
Φ
E = ­N ∆ B ∆t (BA) E = ­N ∆
∆t (BA)f ­ (BA)0
E = ­N
∆t (BfAf) ­ (B0A0)
E = ­N
∆t In this case Af = A0 = A this becomes:
(BfA) ­ (B0A)
E = ­N
∆t Since A is in both terms, it can be factored out
A(Bf ­ B0)
E = ­N
∆t But [Bf ­ B0] is just ∆ B
E = ­NA ∆B
∆t The same approach allows me to factor out B if it is not changing while A is changing
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LENZ’S LAW
Conservation of Energy
An induced emf always gives rise to an induced current whose magnetic field opposes the original change in flux. This accounts for the negative sign in Faraday’s Law.
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Concept Development Lenz's Law
3. If the circular conductor undergoes thermal expansion while in a uniform magnetic field, a current will be induced clockwise around it. Is the magnetic field directed into the page or out of it?
OUT
OUT
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CP 2: The figure shows three situations in which identical circular conducting loops are in uniform magnetic fields that are either increasing (Inc) or decreasing (Dec) in magnitude at identical rates. In each, the dashed line coincides with a diameter. Rank the situations according to the magnitude of the current induced in the loops, greatest first.
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A+B C=0
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CP3: The figure shows four wire loops, with edge lengths of either L or 2L. All four loops will move through a region of uniform magnetic field B (directed out of the page) at the same constant velocity. Rank the four loops according to the maximum magnitude of the emf induced as they move through the field, greatest first.
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C+D, A+b
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2. A long straight wire with current i passes (without touching) three rectangular wire loops with edge lengths L, 1.5L, and 2L. The loops are widely spaced (so as to not affect one another). Loops 1 and 3 are symmetric about the long wire. Rank, greatest first, the loops according to the size of the current induced in them if current i is (a) constant (b) Increasing
a) all tie
b) 2, 1+3 = 0
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4. A circular loop moves at constant velocity through regions where uniform magnetic fields of the same magnitude are directed into or out of the page. (The field is zero outside the dashed lines.) At which of the seven indicated loop positions is the emf induced in the loop (a) Clockwise
(b) Counterclockwise
(c) Zero?
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(a) Clockwise
(b) CCW
(c) Zero?
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(a) Clockwise 2, 6
(b) Counterclockwise 4
(c) Zero? 1, 3, 5, 7
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In each situation, the same uniform magnetic field pierces the loops and is increasing in magnitude at the same rate. Rank the three situations according to the magnitude of the emf induced in the loops, greatest first, if the magnetic field points in the positive direction of: (a) y
(b) Z
(c) X
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(a) y
(b) Z
(c) X
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(a) y all tie =0
(b) Z all tie not = 0
(c) X 3, 1+2 =0
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8. What is the direction of the induced current in the circular loop due to the current shown in each part? a ccw b cw c 0 d ccw
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8. If the variable resistance R in the left hand circuit is increased at a steady rate, is the current induced in the right hand loop clockwise or counterclockwise?
CW V=IR I DECREASE R INCREASE
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9. If the solenoid is being pulled away from the loop shown, in what direction is the induced current in the loop? 55
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9. A circular region, figure a, in which an increasing uniform magnetic field is directed out of the page, as well as a concentric circular path along which E is to be evaluated. The table gives the initial magnitude of the magnetic field, the increase in that magnitude, and the time interval for the increase, in three situations. Rank the situations according to the magnitude of the electric field induced along the path, greatest first.
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10. Figure b shows a circular region in which a decreasing uniform magnetic field is directed out of the page, as well as four concentric circular paths. Rank the paths according to the magnitude of E
evaluated along them, greatest first.
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4. The north pole of the magnet in the figure is being inserted into the coil. In which direction is the induced current flowing through the resistor R?
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7. (a) If the resistance of the resistor in the figure is slowly increased, what is the direction of the current induced in the small circular loop inside the larger loop? (b) What would it be if the small loop were placed outside the larger one, to the left?
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a. R inc I dec B out dec therefore Ii is ccw
b. cw
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GI 3. The rectangular loop shown is pushed into the magnetic field which points inward as shown. In what direction is the induced current? CCW
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Faraday’s Law of Induction; Lenz’s Law
Magnetic flux will change if the angle between the loop and the field changes:
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Faraday Law Problems
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What is the magnitude of the induced emf in a single loop 2.0m 2 coil if it is perpendicular to a 0.50T magnetic field which is turned off over a time interval of 4.0s?
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What is the magnitude of the induced emf in a ten loop coil of wire whose area is 2.0m 2 if it is perpendicular to a magnetic field which is increased from 0.30T to 1.5T over a time interval of 4.0s?
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A 4.0m 2 single loop coil of wire is initially perpendicular to a 0.60T magnetic field. It is then rotated so that it become parallel to that magnetic field 2.0s later. What is the magnitude of the induced emf?
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A coil of wire, consisting of 50 loops, is perpendicular to a 1.2T magnetic field. The area of the coil is increased from 0.40m 2 to 1.2m 2 over a time of 5.0s. What is the magnitude of the induced emf in the coil? 67
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Lenz’s Law
The minus sign tells us that the direction of the induced emf is such that the resulting current produces a magnetic field that resists the change of flux through the loop. For instance, if the external field gets weaker, the current tries to replace the "missing" external field.
If the external field gets stronger, the induced current tries to opposes the "extra" external field.
Only worry about the field within the loop , ignore the field outside it.
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Lenz’s Law
Initial External Field (red)
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Final External Field (red)
Plus Field due to Induced Current (blue)
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. . . .
The current flows CCW to create a field out of the board (dots) to oppose the change in the external field
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Lenz’s Law
Initial External Field (red)
Final External Field (red)
. . . .
. . . . . .
. . . . . .
. . . .
Plus Field due to Induced Current (blue)
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. . . . . .
. . . . . .
. . . .
x x x x
x x x x x x
x x x x x x
x x x x The current flows CW to create a field into the board (blue "x"s) to cancel the new external field (red dots)
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Lenz’s Law
Initial External Field (red)
x x x x
x x x x x x
x x x x x x
x x x x Final External Field (red)
. . . .
. . . . . .
. . . . . .
. . . .
Plus Field due to Induced Current (blue)
. . . .
. . . . . .
. . . . . .
. . . .
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 The current flows CW to create a field into the board (blue "x"s) to cancel the new external field (red dots) and replace the old external field (red "x"s)
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Faraday’s Law of Induction; Lenz’s Law
Problem Solving: Lenz’s Law
Determine whether the magnetic flux is increasing, decreasing, or unchanged.
The magnetic field due to the induced current points in the opposite direction to the original field if the flux is increasing; in the same direction if it is decreasing; and is zero if the flux is not changing.
Use the right­hand rule to determine the direction of the current.
Remember that the external field and the field due to the induced current are different.
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A magnetic field is pointing straight up through a coil of wire. The field is suddenly turned off. What is the direction of the induced current in the wire?
A
Out of the page
B
Into the page
C
Clockwise
D
Counterclockwise
E
There is no induced current
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A magnetic field is pointing straight up through a coil of wire. The field is suddenly doubled in magnitude. What is the direction of the induced current in the wire?
A
Out of the page
B
Into the page
C
Clockwise
D
Counterclockwise
E
There is no induced current
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A coil of wire is sitting on a table top. A magnet is held above it with it's north pole pointed downwards. What is the direction of the induced current?
A
Out of the page
B
Into the page
C
Clockwise
D
Counterclockwise
E
There is no induced current
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A coil of wire is sitting on a table top. A magnet is held above it with it's north pole pointed downwards. The magnet is released and falls towards the coil. What is the direction of the induced current?
A
Out of the page
B
Into the page
C
Clockwise
D
Counterclockwise
E
There is no induced current
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Induced EMF in a Moving Conductor MOVING CONDUCTOR in a MAGNETIC FIELD
This image shows another way the magnetic flux can change:
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EMF Induced in a Moving Conductor
The induced current is in a direction that tends to slow the moving bar – it will take an external force to keep it moving.
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EMF Induced in a Moving Conductor
The induced emf has magnitude
Where:
E is the induced emf, measured in volts, V
B is the magnetic field, measured in Tesla, T
l is the length of the conductor, measured in meters, m
V is the velocity of the conductor moving through the magnetic field, measured in m/s
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EMF Induced in a Moving Conductor
Another perspective
ΣF = ma
Magnetic Force
FB ­ FE = 0
Electric Force
qvB ­ qE = 0
vB = E
but E = V/d
vB = V/d
V/d = vB but d is just l and V is E
= Blv
E
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Rail Gun
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C. D. of MOTIONAL EMF 83
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5. Two circuits in which a conducting bar is slid at the same speed v through the same uniform magnetic field and along a U shaped wire.
The parallel lengths of the wire are separated by 2L in circuit 1 and by L in circuit 2. The current induced in circuit 1 is counterclockwise. (a) Is the direction of the magnetic field into or out of the page?
(b) Is the direction of the current in circuit 2 cw or ccw?
(c) Is the current in circuit 1 larger than, smaller than, or the same as that in circuit 2?
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8. In the following figure, a closed loop moves at a constant speed parallel to a long, straight, current­carrying wire. Is there a current in the loop? If so, is this current circulating clockwise or counterclockwise?
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5. A metallic bar moves to the right in a uniform magnetic field that points out of the page. (a) Is the electric field inside the bar zero? Explain. (b) Is an external force required to keep the bar moving with constant velocity? Explain.
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6. In the following figure, the bar moves to the right with a velocity v in a uniform magnetic field that points out of the page. (a) Is the induced current clockwise or counterclockwise? (b) If the bar moves to the left, will the current reverse?
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6. A conducting rod slides at constant velocity V across an incomplete conducting square, with which it makes electrical contact. The square is in a uniform magnetic field that points directly out of the page. During the slide: (a) is the induced current clockwise, counterclockwise, or does it change from one to the other midway?
(b) Is the current constant, increasing, or increasing and then decreasing?
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Motional EMF Problems
13 What is the voltage between the ends of a 100m long metal rod which is traveling at a velocity of 400 m/s perpendicularly through a 5 x 10­4 T magnetic field?
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Changing Magnetic Flux Produces an Electric Field
A changing magnetic flux induces an electric field; this is a generalization of Faraday’s law. The electric field will exist regardless of whether there are any conductors around.
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Electric Generators
A generator is the opposite of a motor – it transforms mechanical energy into electrical energy. This is an ac generator:
The axle is rotated by an external force such as falling water or steam. The brushes are in constant electrical contact with the slip rings.
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Electric Generators
A dc generator is similar, except that it has a split­ring commutator instead of slip rings.
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Transformers and Transmission of Power
A transformer consists of two coils, either interwoven or linked by an iron core. A changing emf in one induces an emf in the other. The ratio of the emfs is equal to the ratio of the number of turns in each coil:
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Transformers and Transmission of Power
This is a step­up transformer – the emf in the secondary coil is larger than the emf in the primary:
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Transformers and Transmission of Power
Energy must be conserved; therefore, in the absence of losses, the ratio of the currents must be the inverse of the ratio of turns:
Pin = Pout
IinVin = Iout Vout
Iprimary Vprimary = Isecondary Vsecondary
Isecondary Vprimary Iprimary = Vsecondary Isecondary Nprimary =
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Transformers and Transmission of Power
Transformers work only if the current is changing; this is one reason why electricity is transmitted as ac.
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Applications of Induction: Sound Systems, Computer Memory, Seismograph, GFCI
This microphone works by induction; the vibrating membrane induces an emf in the coil
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Applications of Induction: Sound Systems, Computer Memory, Seismograph, GFCI
Differently magnetized areas on an audio tape or disk induce signals in the read/write heads.
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Applications of Induction: Sound Systems, Computer Memory, Seismograph, GFCI
A seismograph has a fixed coil and a magnet hung on a spring (or vice versa), and records the current induced when the earth shakes.
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Applications of Induction: Sound Systems, Computer Memory, Seismograph, GFCI
A ground fault circuit interrupter (GFCI) will interrupt the current to a circuit that has shorted out in a very short time, preventing electrocution.
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Summary of Electromagnetism
• Magnetic flux:
• Changing magnetic flux induces emf:
• Induced emf produces current that opposes original flux change
• Changing magnetic field produces an electric field
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