Chapter 20 Induced Voltages and Inductance

General Physics Inductors & RL Circuits Sections 5–8

General Physics Generators  Alternating Current (AC) and Direct Current (DC) generators – Converts mechanical energy to electrical energy – Consists of a wire loop rotated through a magnetic field by some external means – There are a variety of sources that can supply the energy to rotate the loop These may include falling water, heat by burning coal to produce steam

General Physics AC Generators  As the loop rotates (θ changes), the magnetic flux through the loop changes with time  This induces an emf and a current in the external circuit (toaster)  The ends of the loop are connected to slip rings that rotate with the loop  Connections to the external circuit are made by stationary brushes in contact with the slip rings  The output voltage oscillates between positive and negative polarity  The current is an AC current

General Physics AC Generators – Rotating Loop  Wires BC and AD act as bars moving vertically through the horizontal magnetic field between the N and S poles.  An emf is generated in wires BC and AD  The total emf produced in these 2 wires is ε = 2 B ℓ v  = 2 B ℓ v sin θ  If the loop rotates with a constant angular speed, ω, the emf generated by the rotating loop is ε =2 B ℓ (a / 2) ω sin ωt = B A ω sin ωt  If a coil has N turns, the emf is N times as large ε = N B A ω sin ω t Active Figure: AC GeneratorAC Generator

General Physics DC Generators  Components are essentially the same as that of an AC generator  The major difference is the contacts to the rotating loop are made by a split ring, or commutator  The output voltage always has the same polarity  The current is a DC pulsing current Active Figure: DC GeneratorDC Generator

General Physics Motors  Motors are devices that convert electrical energy (through magnetic forces) into mechanical energy – A motor is a generator run in reverse  A motor can perform useful mechanical work when a shaft connected to its rotating coil is attached to some external device

General Physics Motors and Back emf  As the motor rotates, the magnetic flux through the loop changes with time  This induces a back emf that tends to reduce the current applied to the motor from the external source  When a motor is first turned on, the current is very large because there is no back emf initially  As the coil begins to rotate, the induced back emf opposes the applied voltage  The current in the coil is reduced  The power requirements for starting a motor and for running it under heavy loads are greater than those for running the motor under average loads

General Physics Joseph Henry  1797 – 1878  First director of the Smithsonian  First president of the Academy of Natural Science  First to produce an electric current with a magnetic field  Improved the design of the electro-magnetic and constructed a motor  Discovered self-inductance

General Physics Self-inductance  Self-inductance occurs when the changing flux through a circuit arises from the circuit itself – When the switch is closed, the current increases from zero – As the current increases, the magnetic flux through a loop due to this current also increases – The increasing flux induces an emf that opposes the change in magnetic flux – As the magnitude of the current increases, the rate of increase lessens and the induced emf decreases – This opposing emf results in a gradual increase of the current rather than a sharp increase

General Physics Self-inductance, cont  The self-induced emf is proportional to the time rate of change of the current – L is a proportionality constant called the self-inductance of the circuit or device – The SI unit of self-inductance is the Henry 1 H = 1 (V · s) / A – The negative sign indicates that a changing current induces an emf in opposition to that change – Lenz’s law

General Physics Self-inductance, cont  The inductance of a coil depends on geometric factors  You can determine L from the expression  For a solenoid the inductance is

General Physics Self-Inductance and Lenz’ Law  Consider an increasing current through the inductor  The self-induced emf has a direction so as to oppose the increase in the current  Consider a decreasing current through the inductor  The self-induced emf has an opposite direction so as to oppose the decrease in the current

General Physics Inductor in a Circuit – RL Circuit  When the switch is closed, the current in the RL circuit increases from zero  The increasing current induces an emf in the inductor that opposes the change in the current  As the magnitude of the current increases, the rate of increase lessens and the self-induced emf decreases  When the current reaches its maximum, the rate of change and the self-induced emf become zero  The time constant, , for an RL circuit is the time required for the current in the circuit to reach 63.2% of its final value

General Physics RL Circuit, cont  The time constant depends on R and L  The current at any time can be found by Active Figure: An RL CircuitAn RL Circuit

General Physics Energy Stored in a Magnetic Field  The emf induced by an inductor prevents a battery from establishing an instantaneous current in a circuit  The battery has to do work to produce a current – This work results in energy being stored by the inductor in its magnetic field PE L = ½ L I 2 – Note that this result is similar to the expression for the energy stored by a capacitor in its electric field PE C = ½ C ΔV 2