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