Electromagnetic Induction -...
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Electromagnetic Induction
Transcript of Electromagnetic Induction -...
Electromagnetic Induction
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Motional EMF
Consider a length l of conductor moving to the right in a magnetic field that is into the diagram. Positive charges in the conductor will experience an upward force and negative charges a downward force. The net result is that charges will “pile up” at the two ends of the conductor and create an electric field E. When the force produced by E becomes large enough to balance the magnetic force, the movement of charges will stop and the system will be in equilibrium.
BF qvB= EF qE= B EF F E vB= ⇒ =
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Separating Charge
top bottom ( )V V V E x Bv l∆ = − = ∆ = vlB=E
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Conceptual Question 1The square conductor moves upward through a uniform
magnetic field that is directed out of the diagram.Which of the figures shows the correct distribution of
charges on the conductor?
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Induced Current in a Circuit
The figure shows a conducting wire sliding with speed v along a U-shaped conducting rail. The induced emfEEEE will create a current I around the loop.
vlB=EvlB
IR R
= =E
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Conceptual Question 2
Consider the system shown in the figure. Which description of the induced current is correct?
(a) There is a clockwise current;(b) There is a counterclockwise current;(c) There is no current;
Voltage Between Wing TipsA Boeing-747 jet with a wing span of 60 m is flying horizontally with a speed of 300 m/s over Phoenix. Assume the magnetic field is perpendicular to the velocity and has a magnitude of 50 µT. What is the voltage generated between the wing tips?
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Force and Induction
We have assumed that the sliding conductor moves with a constant speed v. It turns out that a current carrying wire in a magnetic field experiences a force Fmag, so we must supply a counter-force Fpull to make this happen.
2 2
pull mag
vlB vl BF F IlB lB
R R = = = =
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Energy Considerations
2 2 2
pull pull
v l BP F v
R= =
2 2 2 22
dissipated
vlB v l BP I R R
R R = = =
Therefore, the work done in moving the conductor is equal to the energy dissipated in the resistance. Energy is conserved.
Whether the wire is moved to the right or to the left, a force opposing the motion is observed.
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Example: Lighting A BulbThe figure shows a circuit including a 3 V
1.5 W light bulb connected by ideal wires with no resistance. The right wire is pulled with constant speed v through a perpendicular 0.10 T magnetic field.
(a) What speed must the wire have to light the bulb to full brightness?
(b) What force is needed to keep the wire moving at a constant speed?
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The wire in the figure below has mass 2 kg and length 1 m. It maintains good contact with the conducting frame. The resistance of the circuit is 2 Ω. A uniform magnetic field with magnitude 1 T is directed perpendicular to the frame (out of the page).
a) What direction is the induced current?
b) The wire will accelerate downward until the upward force on the wire (due to the interaction of the current with the magnetic field) balances the gravitational force. Neglecting frictional effects, determine the terminal velocity of the wire.
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c) Find the current induced in the wire when the wire is falling at its terminal speed.
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Eddy Currents
Suppose that a rigid square copper loop is between the poles of a magnet. If the loop moves, as long as no conductors are in the field of the magnet there will be no current and no forces. But when one side of the loop enters the magnetic field, a current flow will be induced and a force will be produced. Therefore, a force will be required to pull the loop out of the magnetic field, even though copper is not a magnetic material.
However, if we cut the loop, there will be no force.
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Eddy Currents
Another way of looking at the system is to consider the magnetic field produced by the current in the loop. The current loop is effectively a dipole magnet with a S pole near the N pole of the magnet, and vice versa.
The attractive forces between these poles must be overcome by an external force to pull the loop out of the magnet.
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Eddy Currents
Now consider a sheet of conductor pulled through a magnetic field. There will be induced current, just as with the wire, but there are now no well-defined current paths. As a consequence, two “whirlpools” of current will circulate in the conductor. These are called eddy currents.
A magnetic braking system.
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Eddy Currents
• A conductor movingthrough a magnetic field will have induced currents– The currents will where
the field is changing
– The force from the magnetic field on these currents always work against the velocity
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Eddy Currents• A conductor moving through
a magnetic field will have induced currents
– The currents will where the field is changing
– The force from the magnetic field on these currents always work against the velocity
– Cutting strips into the conductor decreases the size of the eddies and thus the retarding force
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The Basic Definition of Flux
Imagine that you are holding a loop of wire of area Ain front of a fan, and you want to know the volume per second F of air with velocity v flowing through the loop.
When the loop is perpendicular to the air flow (a), Fwill be a maximum, and when the plane of the loop is parallel to the air flow (b), F=0.
Then F = vA cosθθθθ,,,, where θθθθ is the angle between v and the unit vector n normal to the plane of the loop, i.e., F = v⊥⊥⊥⊥A.
^
By analogy, the electric flux is: cose E A EA θ⊥Φ = =
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Magnetic Flux
The number of arrows passing through the loop depends on two factors:(1) The density of arrows, which is proportional to B(2) The effective area Aeff = A cosθ of the loop
We use these ideas to define the magnetic flux:
m eff: cosFlux A B AB θΦ ≡ =2 : 1 weber = 1 Wb = 1 TmFlux units
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The Area Vector
B B AΦ ≡ ⋅
We can make the flux expression a vector equation by defining an area vector: Here, n is the unit vector perpendicular to the plane of the area. If the surface is closed, n points away from the interior. We can use this definition of n for curved surfaces as well as flat ones by defining n as perpendicular to the local surface.
With this definition, the magnetic flux ΦΦΦΦB for a constant B field is:
ˆA An≡
^
^ ^^
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Example: A Circular Loop Rotating in a Magnetic Field
The figure shows a 0.1 m diameter loop rotating in a uniform 0.050 T magnetic field.
What is the magnitude of the flux through the loop when the angle is θ = 00, 300, 600, and 900?
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In 1831, Joseph Henry, a Professor of Mathematics and Natural Philosophy at the Albany Academy in New York, discovered magnetic induction. In July, 1832 he published a paper entitled “On the Production of Currents and Sparks of Electricity from Magnetism” describing his work. Because Henry published after Michael Faraday, his did not receive much credit for this discovery, which actually preceded Faraday’s.
The History of Induction
Joseph Henry(1797-1878)
Michael Faraday(1791-1867)
Michael Faraday's ideas about conservation of energy led him to believe that since an electric current could cause a magnetic field, a magnetic field should be able to produce an electric current. Hedemonstrated this principle of induction in 1831 and published his results immediately. The principle of induction was a landmark in applied science, for it made possible the dynamo, or generator, which produces electricity by mechanical means.
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Faraday’s DiscoveryFaraday had wound two coils around the same iron ring.
He was using a current flow in one coil to produce a magnetic field in the ring, and he hoped that this field would produce a current in the other coil. Like all previous attempts to use a static magnetic field to produce a current, his attempt failed to generate a current.
However, Faraday noticed something strange. In the instant when he closed the switch to start the current flow in the left circuit, the current meter in the right circuit jumped ever so slightly. When he broke the circuit by opening the switch, the meter also jumped, but in the opposite direction. The effect occurred when the current was stopping or starting, but not when the current was steady.
Faraday has invented the picture of lines of force, and he used this to conclude that the current flowed only when lines of force cut through the coil.
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Faraday Investigates Induction
Faraday placed one coil above the other, without the iron ring. Again there was a momentary current when the switch opened or closed.
Faraday replaced the upper coil with a bar magnet. He found that there was a momentary current when the bar magnet was moved in or out of the coil.
Was it necessary to move the magnet? Faraday placed the coil in the field of a permanent magnet. He found that there was a momentary current when the coil was moved.Conclusion: There is a current in the coil if and only if the
magnetic field passing through the coil is changing.
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Lenz’s Law
Heinrich Friedrich Emil Lenz(1804-1865)
In 1834, Heinrich Lenz announced a rule for determining the direction of an induced current, which has come to be known as Lenz’s Law.
Here is the statement of Lenz’s Law:There is an induced current in a closed conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field opposesthe change in the flux.
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Lenz’s Law
If the field of the bar magnet is already inthe loop and the bar magnet is removed, theinduced current is in the direction that triesto keep the field constant.
If the loop is a superconductor, a persistentstanding current is induced in the loop, and thefield remains constant.
Superconductingloop
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Six Induced Current Scenarios
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Example: Lenz’s Law
The switch in the circuit shown has been closed for a long time.What happens to the lower loop when the switch is opened?
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Example: Lenz’s Law
The switch in the circuit shown has been closed for a long time.What happens to the lower loop when the switch is opened?
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Example: Lenz’s Law
The figure shows two solenoids facing each other.When the switch for coil 1 is closed, does the current in coil 2 flow from
right to left or from left to right?
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Example: Lenz’s Law
The figure shows two solenoids facing each other.When the switch for coil 1 is closed, does the current in coil 2 flow from
right to left or from left to right?
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Example: A Rotating LoopA loop of wire is initially in the xy plane in a uniform magnetic field in the x
direction. It is suddenly rotated 900 about the y axis, until it is in the yz plane.In what direction will be the induced current in the loop?
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Example: A Rotating Loop
Initially there is no flux through the coil. After rotation the coil will be threaded by magnetic flux in the x direction. The induced current in the coil will oppose this change by producing a magnetic field that points in the –x direction. Therefore, the induced current will be clockwise, as shown in the figure.
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Faraday’s Law
B B dA BA Bl xΦ = ⋅ = =∫
Bd d dxBl x Bl
dt dt dt
Φ = =
dxBlv Bl
dt= =E
BTherefore, d
dt
Φ=E
This is Faraday’s Law. It can be stated as follows:An emfE is induced in a conducting loop if the magnetic flux ΦΦΦΦB through the
loop changes with time, so that E = |dΦΦΦΦB/dt| for the loop. The emf will be in the direction that will drive the induced current to opposethe flux change, as given by Lenz’s Law.
Consider the loop shown:
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Example: Electromagnetic Induction in a Circular Loop
The magnetic field shown in the figure decreases from 1.0 T to 0.4 T in 1.2 s. A 6.0 cm diameter loop with a resistance of 0.010 Ω is perpendicular to the field.
What is the direction of the current induced in the loop?
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Example: Electromagnetic Induction in a Circular Loop
The magnetic field shown in the figure decreases from 1.0 T to 0.4 T in 1.2 s. A 6.0 cm diameter loop with a resistance of 0.010 Ω is perpendicular to the field.
What is the size of the current induced in the loop?
The current direction is such as to reinforce the diminishing B field. Therefore, the current Iwill be clockwise.
I
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What does Faraday’sLaw Tell Us?
Faraday’s Law tells us that all induced currents are the associated with a changing magnetic flux. There are two fundamentally different ways to change the magnetic flux through a loop:
(1) The loop can move, change size, or rotate, creating motional emf;(2) The magnetic field can change in magnitude or direction.
We can write: Bd dA dBB A
dt dt dt
Φ= = ⋅ + ⋅
E
motionalemf
newphysics
The second term says that an emf can be created simply by changing a magnetic field, even if nothing is moving.
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Generators and Applications of Induced
EMF
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Generators and Motors
• A current loop, rotating in a constant magnetic field generates a varying EMF
tNABdt
dN
tAB
B
B
ωω
ω
ε sin
cos
=Φ−=
=⋅=Φ AB
N Turns in loop, each contribute
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GeneratorsThe figure shows a coil with N
turns rotating in a magnetic field, with the coil connected to an external circuit by slip rings that transmit current independent of rotation. The flux through the coil is:
cos
cosB A B AB
AB t
θω
Φ = ⋅ ==
( )coil cos sinBd dN ABN t ABN t
dt dtω ω ωΦ= = = −E
Therefore, the device produces emf and current that will vary sinusoidally, alternately positive and negative. This is called an alternating current generator, producing what we call AC voltage.
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Example: An AC GeneratorA coil with area 2.0 m2 rotates in a 0.10 T magnetic field at a frequency of 60 Hz.
How many turns are needed to generate an AC emf with a peak voltage of 160 V?
coil sinABN tω ω= −E
max ABNω=E
2 fω π=
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TransformersWhen a coil wound around an
iron core is driven by an AC voltageV1cos ωωωωt, it produces an oscillatingmagnetic field that will induce anemf V2cos ωωωωt in a secondary coilwound on the same core. This iscalled a transformer.
The input emfV1 causes acurrent I1 in the primary coil which produces a magnetic field thatis proportional to 1/N1. The flux inthe iron is proportional to this, andit induces an emfV2 in the secondary coil that is proportional to N2. Therefore, V2 = V1(N2/N1). From conservation of energy, assuming no losses in the core, V1I1 = V2I2. Therefore, the currents in the primary and secondary are related by the relation I1 = I2(N2/N1).
A transformer with N2>>N1 is called a step-up transformer, which boosts the secondary voltage. A transformer with N2<<N1 is called a step-down transformer, and it drops the secondary voltage.
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The Tesla CoilA special case of a step-up
transformer is the Tesla coil. It uses no magnetic material, but has a very high N2/N1 ratio and uses high-frequency electrical current to induce very high voltages and very high frequencies in the secondary.
There is a phenomenon called “the skin effect” that causes high frequency AC currents to reside mainly on the outer surfaces of conductors. Because of the skin effect, one does not feel (much) the electrical discharges from a Tesla coil.
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Metal DetectorsMetal detectors like those used at airports can detect any metal objects, not just
magnetic materials like iron. They operate by induced currents.A transmitter coil sends high frequency alternating currents that will induce current
flow in conductors in its field. Because of Lenz’s Law, the induced current opposes the field from the transmitter, so that net field is reduced. A receiver coil detects the reduction in the magnetic fields from the transmitter and registers the presence of metal.
