Motional emf Motional emf is the voltage induced across a conductor moving through a magnetic field. If a metal rod of length L moves at velocity v through a magnetic field B, the motional emf is: ε = vlb as long as the velocity, field, and length are mutually perpendicular. In which direction do positive charges deflect, if they move with a metal rod to the right in a magnetic field directed into the page?
Motional emf In which direction do positive charges deflect, if they move with a metal rod to the right in a magnetic field directed into the page? F = qvbsinθ Use the right-hand rule associated with. Positive charges deflect up, and negative charges deflect down.
Acting like a battery The moving rod can act like a battery if we connect it up in a circuit, like so. The rod is placed on a pair of conducting rails that are separated by a distance L. The rails are connected at the left end by a resistor of resistance R - assume the resistance of the rod and rails is negligible compared to R. There is a uniform magnetic field of magnitude B directed into the page.
Direction of the induced current? If the rod (in red) is moved to the right, will there be an induced current? If so, in what direction is it? 1. Clockwise 2. Counterclockwise 3. There is no induced current
Apply the pictorial method Before After The simulation draws the Before and After pictures for us. To oppose the change, the loop needs to create field lines out of the page, requiring a counterclockwise induced current.
Acting like a battery The rod is initially at rest, but is then subjected to a constant force F directed right. Neglect friction between the rod and the rails. What happens? 1. The bar experiences a constant acceleration, and the speed increases at a constant rate 2. The changing flux gives rise to another force in the same direction as F that accelerates the bar even faster than F would by itself. 3. The changing flux gives rise to another force opposite in direction to F that causes the bar to reach a terminal (constant) velocity 4. The changing flux gives rise to another force opposite in direction to F that causes the bar to come to rest
Acting like a battery ILB F The faster the rod goes, the larger the current induced in the loop consisting of rod, rail, resistor, rail. This current gives rise to a force of magnitude ILB opposing the applied force F. When the forces are equal and opposite, there is no net force, so the rod continues to move at a constant velocity.
Eddy currents An eddy current is a swirling current set up in a conductor in response to a changing magnetic field. By Lenz's law, the swirling current sets up a magnetic field opposing the change. In a conductor, electrons swirl in a plane perpendicular to the magnetic field. Eddy currents cause energy to be lost. More accurately, eddy currents transform more useful forms of energy, such as kinetic energy, into heat, which is generally much less useful. In many applications the loss of useful energy is not particularly desirable, but there are some practical applications such as train brakes.
Eddy current application: train brakes During braking, the metal wheels are exposed to a magnetic field from an electromagnet, generating eddy currents in the wheels. The magnetic interaction between the applied field and the eddy currents acts to slow the wheels down. The faster the wheels are spinning, the stronger the effect, meaning that as the train slows the braking force is reduced, producing a smooth stopping motion. Draw a set of pictures for the green region, to show the direction of the induced current in the green region as it enters the magnetic field.
Eddy current application: train brakes During braking, the metal wheels are exposed to a magnetic field from an electromagnet, generating eddy currents in the wheels. The magnetic interaction between the applied field and the eddy currents acts to slow the wheels down. The faster the wheels are spinning, the stronger the effect, meaning that as the train slows the braking force is reduced, producing a smooth stopping motion. In which direction is the force on this induced current?
Eddy current application: train brakes During braking, the metal wheels are exposed to a magnetic field from an electromagnet, generating eddy currents in the wheels. The magnetic interaction between the applied field and the eddy currents acts to slow the wheels down. The faster the wheels are spinning, the stronger the effect, meaning that as the train slows the braking force is reduced, producing a smooth stopping motion.
Eddy current question To stop (or slow down) the train, an electromagnet is turned on, passing a magnetic field through sections of the train's wheels. The eddy currents set up in the wheels act to slow the train down. What would happen if the direction of the magnetic field was reversed? 1. The train would still slow down. 2. The train would speed up.
Eddy current question If the field goes the other way, the eddy currents also reverse direction. Reversing both the field and the current gives a force (F = ILB) in the same direction the train still slows down.
Electric generators An electric motor transforms electrical energy into mechanical energy. An electric generator transforms mechanical energy into electrical energy. That is, it generates electricity. The same device a coil in a magnetic field can be used as a motor or a generator.
Electric generators If a current is passed through the coil, the interaction of the magnetic field with the current causes the coil to spin that s a motor. If we spin the coil, the changing flux through the coil induces a current now it s a generator.
Direction of the induced current? If the loop spins so the magnetic flux through the loop decreases, in what direction is the induced current in the loop? 1. Clockwise 2. Counterclockwise 3. There is no induced current
Apply the pictorial method Before After To Oppose
Apply the pictorial method Before After To Oppose
Maximum current? At what instant is the magnitude of the current maximum? 1. When the plane of the loop is perpendicular to the field (maximum area) 2. When the plane of the loop is parallel to the field (zero area) 3. Because the loop is spinning at a constant rate, the magnitude of the current is constant
Maximum current The magnitude of the current is proportional to the slope of the flux graph, so let s draw the flux graph.
Maximum current The magnitude of the current is proportional to the slope of the flux graph, so let s draw the flux graph. When is the current maximum?
Maximum current The magnitude of the current is proportional to the slope of the flux graph, so let s draw the flux graph. When is the current maximum? When the flux is zero, where the slope of the flux graph is largest.
Electric generators Let's say we spin a coil of N turns and area A at a constant rate in a uniform magnetic field B. By Faraday's law, the induced emf is given by: B and A are constants, and if the angular speed ω of the loop is constant the angle is: θ = ω t. The induced emf is: ε = N ( BAcos θ ) t (cos ωt) ε = NBA = ωnbasin( ωt) t Spinning a loop in a magnetic field at a constant rate is an easy way to generate AC electricity. The peak voltage is: ε max = ωnba
Back emf When something like a refrigerator or an air conditioner (anything with a motor) first turns on in your house, the lights often dim momentarily. The lights brighten again because a spinning motor acts like a generator. This emf generated, known as the back emf, acts against the applied voltage that's causing the motor to spin, and reduces the current flowing through the coils of the motor. At the motor's operating speed, enough current flows to overcome any losses due to friction and other sources and to provide the necessary energy required for the motor to do work. This is generally much less current than is required to get the motor spinning in the first place.
Back emf If the applied voltage is V, then the initial current flowing through a motor with coils of resistance R is: I = V R For example: 120 V I = = 20 A 6 Ω A device drawing that much current reduces the voltage provided to other devices in your house, causing lights to dim. When the motor is spinning and generating a back emf ε, the current is reduced to: I ( V ε ) R = For example: 120 V 108 V = = 2 A 6 Ω It takes very little time for the motor to reach operating speed and for the current to drop from its high initial value. This is why the lights dim only briefly. I
Transformers A transformer is a device for changing the voltage of an AC signal. A transformer has two coils linked by a ferromagnetic core so the magnetic flux from one passes through the other. When the flux generated by one coil changes (as it does continually if the coil is connected to an AC power source), the flux passing through the other will change, inducing a voltage in the second coil. With AC power, the voltage induced in the second coil will also be AC.
Transformers Both coils are exposed to the same changing flux, so: Φ Vp V = = t N N p Energy (or, equivalently, power) has to be conserved, so: VI = VI p p s s s s N V I = = N V I p p s s s p
DC transformer The primary coil (with N p turns) of a transformer is connected to a battery. The secondary coil (with N s turns) is connected to a resistor. When the switch on the primary side is closed, what is the current through the resistor? 1. zero - there is no current 2. constant and equal to N p / N s times the current in the primary 3. constant and equal to N s / N p times the current in the primary 4. there is a brief current and then the current drops to zero
DC transformer, II The switch is now moved to the secondary side. When the switch is closed, what is the current through the resistor? 1. zero - there is no current 2. constant and equal to N p / N s times the current in the primary 3. constant and equal to N s / N p times the current in the primary 4. there is a brief current and then the current drops to zero
Power transmission Electricity is often generated a long way from where it is used, and is transmitted long distances through power lines. Although the resistance of a short length of power line is relatively low, over a long distance the resistance can become substantial. A power line of resistance R causes a power loss of I 2 R ; this is wasted as heat. By reducing the current, therefore, the I 2 R losses can be minimized. Power companies use step-up transformers to boost the voltage to hundreds of kv before it is transmitted down a power line, reducing the current and minimizing the power lost in transmission lines. Step-down transformers are used at the other end, to decrease the voltage to the 120 V used in household circuits.