PHYSICS 1B Today s lecture: Generators Eddy Currents Self Inductance Energy Stored in a Magnetic Field
PHYSICS 1B Lenz's Law Generators Electric generators take in energy by work and transfer it out by electrical transmission. The AC generator consists of a loop of wire rotated by some external means in a magnetic field. Lenz's Law & Generators 27/08/2013
PHYSICS 1B Lenz's Law A Rotating Loop Assume a loop with N turns, all of the same area, rotating in a magnetic field. The flux through one loop at any time, t, (using the fact that θ = ωt ): Φ B = BA cos θ = BA cos ωt Therefore, the induced emf of the loop is: ε = N dφ B dt = NBA d (cos ωt) dt i.e. ε = NABω sin ωt Lenz's Law & Generators 27/08/2013
PHYSICS 1B Lenz's Law A Rotating Loop ε = NABω sin ωt The relationship is sinusoidal, with a maximum value, ε max given by: ε max = NABω ε max occurs when ωt = 90⁰ or 270 0 i.e. it occurs when the magnetic field is in the plane of the coil, and the rate of change of the flux is at a maximum. Lenz's Law & Generators 27/08/2013
PHYSICS 1B Lenz's Law DC Generators The DC (direct current) generator has essentially the same components as the AC generator. The main difference is that the contacts to the rotating loop are made using a split ring called a commutator. Lenz's Law & Generators 27/08/2013
PHYSICS 1B Lenz's Law DC Generators In this configuration, the output voltage always has the same polarity. It pulsates with time, as was the case with the AC generator. To obtain a steady DC current, commercial generators use many coils and commutators distributed so the pulses are out of phase. Lenz's Law & Generators 27/08/2013
PHYSICS 1B Lenz's Law Motors Motors are devices into which energy is transferred by electrical transmission while energy is transferred out by work. In other words, a motor is a generator operating in reverse. A current is supplied to the coil by a battery, and the torque acting on the current-carrying coil causes it to rotate. Useful mechanical work can be done by attaching the rotating coil to some external device. Lenz's Law & Generators 27/08/2013
PHYSICS 1B Lenz's Law Motors However As the coil rotates in a magnetic field, an emf is induced. This induced emf always acts to reduce the current in the coil. This is called a back emf The back emf increases in magnitude as the rotational speed of the coil increases. Lenz's Law & Generators 27/08/2013
PHYSICS 1B Lenz's Law Motors The current in the rotating coil is therefore limited by the back emf. The term back emf is commonly used to indicate an emf that tends to reduce the supplied current. The induced emf explains why the power requirements for starting a motor and for running it are greater for heavy loads than for light ones. Lenz's Law & Generators 27/08/2013
PHYSICS 1B Lenz's Law Hybrid Drive Systems In a car with a hybrid drive system, a petrol engine and an electric motor are combined to increase the fuel economy of the vehicle and reduce its emissions. Power to the wheels can come from the petrol engine or the electric motor. In normal driving, the electric motor accelerates the vehicle from rest to about 15 mph. During these acceleration periods, the engine isn t running, so petrol is not used, and there is no emission. At higher speeds, the motor and engine work together so that the engine is always operating at, or near, its most efficient speed. This results in a greatly improved fuel efficiency than that of a traditional car. Lenz's Law & Generators 27/08/2013
PHYSICS 1B Lenz's Law Quick Quiz In an AC generator, a coil with N turns of wire spins in a magnetic field. Of the following choices, which will not cause an increase in the emf generated in the coil? a) Replacing the coil wire with one of lower resistance b) Spinning the coil faster c) Increasing the magnetic field d) Increasing the number of turns of wire on the coil Lenz's Law & Generators 27/08/2013
PHYSICS 1B Lenz's Law Quick Answer In an AC generator, a coil with N turns of wire spins in a magnetic field. Of the following choices, which will not cause an increase in the emf generated in the coil? a) Replacing the coil wire with one of lower resistance While reducing the resistance may increase the current that the generator provides to a load, it does not alter the emf. Since ε = NABω sin ωt, the emf depends on ω, B, and N, and so all the other choices increase the emf. Lenz's Law & Generators 27/08/2013
PHYSICS 1B Lenz's Law Eddy Currents Circulating currents called eddy currents are induced in bulk pieces of metal moving through a magnetic field. From Lenz s law, their direction is to oppose the change that causes them. The eddy currents are in opposite directions as the plate enters or leaves the field. Eddy currents are often undesirable because they represent a transformation of mechanical energy into internal energy. Lenz's Law & Generators
PHYSICS 1B Lenz's Law Eddy Currents - Example The magnetic field here is directed into the screen. The induced eddy current is counter-clockwise as the plate enters the field. It is opposite when the plate leaves the field (clockwise). The induced eddy currents produce a magnetic retarding force, and the swinging plate eventually comes to rest. Lenz's Law & Generators
PHYSICS 1B Lenz's Law Eddy Currents - Final There are a couple of ways to reduce the energy loses resulting from eddy currents. The conducting parts can: Be built up in thin layers separated by a nonconducting material Have slots cut in the conducting plate Both these tricks prevent large current loops from forming, and therefore increase the efficiency of the device. Lenz's Law & Generators
PHYSICS 1B Lenz's Law Quick Quiz In equal-arm balances from the early twentieth century, it is sometimes observed that an aluminium sheet hangs from one of the arms and passes between the poles of a magnet. This causes the oscillations of the equal-arm balance to decay rapidly, when otherwise (without this magnetic braking), the oscillation might continue for a long time. The oscillations decay because: a) The aluminium sheet is attracted to the magnet b) Currents in the aluminium sheet set up a magnetic field that opposes the oscillations a) Aluminium is paramagnetic Lenz's Law & Generators
PHYSICS 1B Lenz's Law Quick Answer The oscillations decay because: b) Currents in the aluminium sheet set up a magnetic field that opposes the oscillations When the aluminium sheet moves between the poles of the magnet, eddy currents are established in the aluminium. According to Lenz s law, these currents are in a direction so as to oppose the original change, which is the movement of the aluminium sheet in the magnetic field. Lenz's Law & Generators
PHYSICS 1B Lenz's Law Applied Eddy Currents Eddy currents aren t always a bad thing. In fact, they re commonly used to look for unseen cracks in structures, such as the wings of aircraft. Eddy currents are induced in the structure. When the eddy current falls off, then a crack is probably present, stopping the current! Lenz's Law & Generators
PHYSICS 1B Lenz's Law Charging by induction Induced emf is used to wirelessly recharge things. An alternating current in the base induces an emf in the device being charged. The alternating current thus created is used to charge the device s battery. The big advantage is there are no exposed wires! So no corrosion of the parts, and no chance of an electric shock in the wet. But this is slower and less efficient This is very handy from a health and safety point of view. Lenz's Law & Generators
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Self-Inductance When the switch is closed, the current does not immediately reach its maximum value of ε R due to self-inductance. There is also a self-induced emf, ε L We use Faraday s law to describe the effect.
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Self-Inductance As the current increases with time, the magnetic flux induced through the circuit also increases with time. This induces an emf in the circuit. The direction of the induced emf is to cause an induced current which establishes a magnetic field opposing the change in the field. i.e. opposing the direction of emf in the battery The result is a gradual increase in the current to its final equilibrium value.
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Self-Inductance So a time-varying current in a circuit produces an induced emf opposing the emf that initially set up the time-varying current. This is the basis of the electrical circuit element called an inductor. Energy is stored in the magnetic field of an inductor. There is an energy density associated with the magnetic field.
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Self-Inductance in a Coil a) A current in the coil produces a magnetic field directed towards the left b) If the current increases, the increasing flux causes an induced emf c) The polarity of the induced emf reverses if the current decreases
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Equations for Self-Inductance The induced emf is proportional to the rate of change of the current: ε L = L di dt Here, L is a constant of proportionality called the inductance of the coil. The inductance of a coil depends on the geometry of the coil and its physical characteristics. The unit of inductance is the henry, H. 1H = 1 V s A
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Joseph Henry 1797 1878 American physicist. First director of the Smithsonian First president of the Academy of Natural Science Improved the design of the electromagnet Constructed one of the first motors Discovered self-inductance, but didn t publish his results! Despite this, the unit of inductance is named in his honour
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Terminology We use emf and current when they are caused by batteries or other sources. We use induced emf and induced current when they are caused by changing magnetic fields. When dealing with problems in electromagnetism, it is important to distinguish between the two situations
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Inductance of a Coil Say we have a closely spaced coil, with N turns, through which a current I is flowing. From Faraday s law, we know: ε L = N dφ B dt We also know that ε L = L di dt, so subbing this in L di dt = N dφ B dt LdI = NdΦ B L = NΦ B I = ε L di dt
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Inductance of a Coil L = NΦ B I = ε L di dt The inductance is therefore a measure of the opposition to a change in the current. This can be compared to the resistance, which is a measure of the opposition to the current itself
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Inductance of a Solenoid Now consider a uniformly wound solenoid which has N turns and length l. We then have: We know Φ B = BA, so therefore: B = μ 0 ni = μ 0 N l I Φ B = BA = μ 0 NA l I Therefore: L = NΦ B I = μ 0N 2 A l Note how the inductance just depends on the geometry just like capacitance!
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Quick Quiz The circuit shown in the figure consists of a resistor, an inductor, and an ideal battery that has no internal resistance. At the instant just after the switch is closed, across which circuit element is the voltage equal to the emf of the battery? a) The resistor b) The inductor c) Both the inductor and resistor
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Quick Answer The circuit shown in the figure consists of a resistor, an inductor, and an ideal battery that has no internal resistance. At the instant just after the switch is closed, across which circuit element is the voltage equal to the emf of the battery? b) The inductor As the switch is closed, there is no current, so there is no voltage across the resistor (remember, V = IR )
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Quick Quiz After a very long time, across which circuit element is the voltage equal to the emf of the battery? a) The resistor b) The inductor c) Both the inductor and resistor
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Quick Answer After a very long time, across which circuit element is the voltage equal to the emf of the battery? a) The resistor After a long time, the current has reached its final value, and the inductor has no further effect on the circuit
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Quick Quiz A coil with zero resistance has its ends labelled a and b. The potential of the induced emf at a is higher than at b. Which of the following could be consistent with this situation? a) The current is constant and directed from a to b b) The current is constant and directed from b to a c) The current is increasing and is directed from a to b d) The current is decreasing and is directed from a to b e) The current is increasing and is directed from b to a f) The current is decreasing and is directed from b to a
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Quick Answer A coil with zero resistance has its ends labelled a and b. The potential of the induced emf at a is higher than at b. Which of the following could be consistent with this situation? d) The current is decreasing and is directed from a to b e) The current is increasing and is directed from b to a For the constant current in a) and b), there is no potential difference across the resistance-less inductor. In e), if the current increases, the emf induced in the inductor is in the opposite direction, therefore from a to b, making a higher in potential than b. Similarly, in d), the decreasing current induces an emf in the same direction as the current, from a to b, again making the potential higher at a than b
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Energy in a Magnetic Field In a circuit with an inductor, the battery must supply more energy than in a circuit without an inductor. Part of the energy supplied by the battery appears as internal energy in the resistor. The remaining energy is stored in the magnetic field of the inductor.
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Energy in a Magnetic Field The power supplied by the battery is equal to the dissipation across the resistor, of resistance R, plus the rate at which energy is stored in the inductor, of inductance L. Therefore: In this equation: εi = I 2 R + LI di dt εi is the rate at which energy is supplied by the battery I 2 R is the rate at which the energy is being delivered to the resistor, So, therefore, LI di magnetic field dt must be the rate at which energy is being stored in the
PHYSICS 1B Self Inductance and Energy Stored in a Magnetic Field Energy in a Magnetic Field If we let U denote the energy stored in the inductor at any time, then the rate at which the energy is stored is: du dt = LI di dt To find the total energy, we therefore integrate: U = L I IdI 0 = 1 2 LI2 (compare this result to that for electric fields in a capacitor: U E = 1 2 Q 2 C )