Chapter 20

1. Chapter 20 Induced Voltages and Inductance

2. Inductors & RL Circuits Sections 5–8

3. 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

4. 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

5. 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 Generator

6. 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 Generator

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. RL Circuit, cont • The time constant depends on R and L • The current at any time can be found by Active Figure: An RL Circuit

16. 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 PEL = ½ L I2 • Note that this result is similar to the expression for the energy stored by a capacitor in its electric field PEC = ½ C ΔV2