Transcript
ELECTROMAGNETIC INDUCTION
Induced emf and Induced Current
When a magnet moves relative to a coil, or if the coil moves relative to the magnet, an emf is generated. The fact that the coil is in a changing magnetic field is what causes the emf to appear.
As you can see from the drawing, the motion of the magnet causes a change in the magnetic field strength and therefore the magnetic flux through the coil.
This emf that is generated by the changing flux is called an induced emf and results in an induced current.
Another way to induce a current in a coil is to keep both the coil and the magnet stationary but change the area of the coil. This changes the magnetic flux through the coil resulting in an induced emf.
A third possibility is to keep two coils stationary near each other and vary the current in one of them. This causes a changing magnetic field to exist in the other coil and generates an emf. This effect is used in transformers.
Electromagnetic induction is the process of producing an emf and the resulting current through the interaction of a conductor and a magnetic field.
Motional emf
When a conductor moves in a magnetic field, the charges in the conductor experience a force given by the equation F = qvBsin? where ? is the angle between the velocity vector and the magnetic field vector.
This force causes negative charge to move to one end of the conductor leaving a net positive charge at the other end.
This separation of charge results in an induced emf called a motional emf since it exists because of the motion of charges through a magnetic field.
An external circuit with a device like a light bulb can be connected and powered as long as the rod moves.
The magnitude of the emf generated depends on the length of the wire, the velocity of the wire and the magnetic field strength. If they are all perpendicular to each other, the equation is:
emf = vBL
Example
The drawing shows a type of flow meter that can be used to measure the speed of blood in a blood vessel when it is exposed. Blood can be treated as a moving conductor. Electrodes can be used to measure the small voltage that develops across the vessel. If the vessel has a diameter of 5.6 mm and is placed in a 0.60T magnetic field, find the blood velocity that results in a 1.5 mV reading.
When a current exists in a wire moving in a magnetic field, another force is generated by the interaction of the current and the magnetic field. The equation for the force is F = ILBsin? where ? is the angle between the magnetic field vector and the direction of the current.
If we use the right hand rule to find the direction of the force, we see that it opposes the velocity of the wire.
This means that the moving wire will come to rest unless a force acts to the right to put the wire in equilibrium.
This means that the electrical energy used by the light bulb comes from the kinetic energy of the wire which is supplied by the work done by the force which keeps it moving.
This is consistent with the principle of conservation of energy.
In the drawing below, three identical rods, A, B, and C move in different planes. A constant magnetic field of 0.45 T is directed along the positive y axis. The length of each rod is 1.3 meters and the speeds are all 2.7 m/s. Find the magnitude of the motional emf for each rod and determine which end of each rod is positive.
Magnetic Flux
Magnetic flux is defined the same way as electric flux, that is, the field strength multiplied by the area through which it passes. The difference is that we use magnetic field strength for magnetic flux.
The equation is:
? = BA
The motional emf can then be calculated in terms of changing magnetic flux.
emf = vBL
emf = (x - x0)BL
(t - t0)
emf = (xL - x0L)B
(t - t0)
emf = (A - A0)B
(t - t0)
emf = (BA - BA0)
(t - t0)
emf = (? - ?0) = ??/?t
(t - t0)
This means that the emf is equal to the time rate of change of the magnetic flux.
Only the component of the magnetic field perpendicular to the surface experiencing the flux actually passes through the surface. This component generates the flux and the equation must include the cosine of the angle between the magnetic field vector and the normal to the surface.
? = Bacos?
Example
A loop of wire bent in the shape of a semicircle with radius 0.20 m is rotated in a constant magnetic field of 0.75 T. The normal to the plane of the loop is initially parallel to the direction of the magnetic field and it is rotated through half of a revolution. Find the change in magnetic flux.
Magnetic flux is sometimes expressed in webers(Wb) which has the dimensions of 1 Tm2. In the diagram below, the number of webers is 3 times as much in figure (a) since the magnetic field strength is 3 times as much through the same area.
Faraday's Law of Electromagnetic Induction
Faraday's Law states that a change in magnetic flux per unit of time causes an emf to exist in the coil that experiences this change. The magnitude of the emf is directly proportional to the time rate of change of flux and the number of turns in the coil. The equation is:
emf = -N(? - ?0)/(t - t0)
emf = -N(??)/?t
Example
During an MRI, the patient is placed in a strong magnetic field. One safety concern is the production of a relatively large induced current in the patient during an equipment failure which results in a rapidly disappearing magnetic field. Suppose the largest body surface through which the magnetic flux passes has an area of 0.032 m2 and its normal is parallel to a magnetic field of 1.5 T. Determine the smallest time period over which the field can vanish if the maximum induced emf is to be 0.010 V.
Lenz's Law
The induced emf caused by a changing magnetic flux has a polarity that leads to an induced current whose direction is such that the induced magnetic field opposes the original flux change.
The bar magnet moving to the right causes an increase in magnetic flux in the loop since the field strength relative to the loop is increasing.
The direction of the current in the loop shown above must be from B to A to produce an induced magnetic field that opposes the original flux change. The resulting induced magnetic field causes a reduction in field strength and therefore a reduction in flux.
Remember that the induced field opposes the change in flux, not the flux itself. When the flux is decreasing, the induced current will cause an induced magnetic field in the same direction as the flux.
Lenz's Law is a result of the law of conservation of energy. If Lenz's law was reversed, so that the induced field increased the rate of change of magnetic flux, induced voltages and currents would provide more energy than is available due to the motion of the system.
Electromagnetic induction
Electric Generators
Electric generators convert energy of motion into electricity by rotating a conducting coil in a magnetic field. The magnitude of the emf produced depends on the number of turns in the coil, the magnetic field strength, the area of the coil, and the rate at which the coil turns.
emf = NAB?sin?t
Since NAB? gives us the maximum voltage, we can write the equation as:
emf = emf0sin2?ft
in which emf0 is the maximum voltage.
A generator that produces a voltage that varies in this manner is called an AC generator. The graph of the emf looks like this.
Example
The coil of the device with the voltage output pictured above has a cross sectional area of 0.020 m2 and contains 150 turns. Find (a) the frequency of the generator in Hz, (b) the angular speed in rad/sec and (c) the magnitude of the magnetic field.
? = 2?f
An electric motor operates on the same principles as a generator. The difference is that the motor converts electrical energy into mechanical energy plus some heat energy due to friction.
Since the construction is nearly identical, the motor acts like a generator as it turns to produce a voltage that opposes the voltage supplied to it.
This opposing voltage is called a back emf and will nearly equal the supplied voltage while the motor is operating at full speed.
The current flowing through the motor is large initially but becomes relatively small when the back emf reaches its full value.
Mutual Inductance and Self Inductance
Mutual induction is the process of generating an emf in a second coil due to a changing current in a primary coil.
Mutual inductance is a proportionality constant for two coils determined by the number of turns in the secondary and magnetic flux in the secondary and inversely proportional to the current in the primary.
M = Ns?s/Ip
This allows us to derive from Faraday's Law the expression for the emf induced in the secondary.
emfs = -M(?Ip/?t)
This means the emf induced in the secondary is proportional to the mutual inductance and the time rate of change of current in the primary.
The unit for inductance is called the Henry and is equal to 1 V·s/A.
Example
The average emf induced in the secondary coil is 0.12 V when the current in the primary changes from 3.4 to 1.6 A in 0.14 s. Find the mutual inductance of the coils.
When there is only one coil present with a changing electric current through it, the changing magnetic field produces an emf in the coil itself. This is called self induction.
The self inductance of a coil is proportional to the number of turns and the net flux through one turn and inversely proportional to the current passing through the coil.
L = N?/I
emf = -L(?I/?t)
The equation for the emf comes from Faraday's Law that states the emf generated depends on the time rate of change of magnetic flux.
Example
The current through a 3.2 mH inductor varies with time according to the graph. What is the average induced emf during each of the three time intervals?
Transformers
A transformer is a device used to increase or decrease AC voltage. It will not work with DC voltage since the emf induced in the secondary coil depends on changing magnetic flux.
Two coils of wire are wound on an iron core. The primary coil is connected to the original source of current. Changing current in the primary coil causes a change in magnetic flux through the secondary coil producing an induced emf in the secondary coil.
Since the same rate of flux change is present in both coils, the voltage in both coils is related to the number of turns in each coil.
Vs/Vp = Ns/Np
If the secondary has more turns than the primary, the transformer is called a step-up transformer since it has a higher voltage output than input.
If the voltage output is lower than the voltage input, it is called a step-down transformer.
Example
A neon sign requires 12,000 V to operate. It gets power from a 220 V outlet. Is a step-up or step-down transformer needed? What is the turns ratio Ns/Np of the transformer?
If a transformer is 100% efficient, the power in the secondary must equal the power in the primary. Most are less than 100% efficient with some of the energy being converted to heat. For a 100% efficient transformer,
Ps = Pp
VsIs = VpIp
Is/Ip = Vp/Vs = Np/Ns
This tells us that the current in a transformer coil is inversely proportional to the number of turns in the coil. If the voltage increases, the current must decrease and vice-versa.
Example
A step-up transformer provides the voltage to operate an electrostatic air filter and has a turns ratio of 50 to 1. The primary is connected to a 120 V outlet and the current in the secondary is 0.0017 A. Find the power consumed by the filter.
