Understanding Self-Induction in RL Circuits and Magnetic Fields

1. 23.5 Self-Induction • When the switch is closed, the current does not immediately reach its maximum value • Faraday’s Law can be used to describe the effect

2. Self-Induction, • As the current increases with time, the magnetic flux through the circuit loop also increases with time • This increasing flux creates an induced emf in the circuit • The direction of the induced emf is opposite to that of the emf of the battery • The induced emf causes a current which would establish a magnetic field opposing the change in the original magnetic field

3. Equation for Self-Induction • This effect is called self-inductance and the self-induced emf eLis always proportional to the time rate of change of the current • L is a constant of proportionality called the inductance of the coil • It depends on the geometry of the coil and other physical characteristics

4. Inductance Units • The SI unit of inductance is a Henry (H) • Named for Joseph Henry • 1797 – 1878 • Improved the design of the electromagnet • Constructed one of the first motors • Discovered the phenomena of self-inductance

5. Inductance of a Solenoid having N turns and Length l The interior magnetic field is The magnetic flux through each turn is The inductance is This shows that L depends on the geometry of the object

6. 23.6 RL Circuit, Introduction • A circuit element that has a large self-inductance is called an inductor • The circuit symbol is • We assume the self-inductance of the rest of the circuit is negligible compared to the inductor • However, even without a coil, a circuit will have some self-inductance

7. RL Circuit, Analysis • An RL circuit contains an inductor and a resistor • When the switch is closed (at time t=0), the current begins to increase • At the same time, a back emf is induced in the inductor that opposes the original increasing current

8. The current in RL Circuit • Applying Kirchhoff’s Loop Rule to the previous circuit gives • The current • where t = L / R is the time required for the current to reach 63.2% of its maximum value

9. RL Circuit, Current-Time Graph • The equilibrium value of the current is e/R and is reached as t approaches infinity • The current initially increases very rapidly • The current then gradually approaches the equilibrium value

10. RL Circuit, Analysis, Final • The inductor affects the current exponentially • The current does not instantly increase to its final equilibrium value • If there is no inductor, the exponential term goes to zero and the current would instantaneously reach its maximum value as expected

11. Open the RL Circuit, Current-Time Graph • The time rate of change of the current is a maximum at t = 0 • It falls off exponentially as t approaches infinity • In general,

14. 23.7 Energy stored 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

15. Energy in a Magnetic Field • Looking at this energy (in terms of rate) • Ie is the rate at which energy is being supplied by the battery • I2R is the rate at which the energy is being delivered to the resistor • Therefore, LI dI/dt must be the rate at which the energy is being delivered to the inductor

16. Energy in a Magnetic Field • Let U denote the energy stored in the inductor at any time • The rate at which the energy is stored is • To find the total energy, integrate and UB = ½ L I2

17. Energy Density in a Magnetic Field • Given U = ½ L I2, • Since Al is the volume of the solenoid, the magnetic energy density, uB is • This applies to any region in which a magnetic field exists • not just in the solenoid

21. Inductance Example – Coaxial Cable • Calculate L and energy for the cable • The total flux is • Therefore, L is • The total energy is

23. 23.8 Magnetic Levitation – Repulsive Model • A second major model for magnetic levitation is the EDS (electrodynamic system) model • The system uses superconducting magnets • This results in improved energy effieciency

24. Magnetic Levitation – Repulsive Model, 2 • The vehicle carries a magnet • As the magnet passes over a metal plate that runs along the center of the track, currents are induced in the plate • The result is a repulsive force • This force tends to lift the vehicle • There is a large amount of metal required • Makes it very expensive

25. Japan’s Maglev Vehicle • The current is induced by magnets passing by coils located on the side of the railway chamber

26. EDS Advantages • Includes a natural stabilizing feature • If the vehicle drops, the repulsion becomes stronger, pushing the vehicle back up • If the vehicle rises, the force decreases and it drops back down • Larger separation than EMS • About 10 cm compared to 10 mm

27. EDS Disadvantages • Levitation only exists while the train is in motion • Depends on a change in the magnetic flux • Must include landing wheels for stopping and starting • The induced currents produce a drag force as well as a lift force • High speeds minimize the drag • Significant drag at low speeds must be overcome every time the vehicle starts up

28. Exercises of Chapter 23 • 5, 9, 12, 21, 25, 32, 35, 39, 42, 47, 52, 59, 65, 67