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 Lis 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

L

dIL

dt

4

Inductance Units The SI unit of inductance is a Henry (H)

AsV

1H1

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

B o

NABA I

o o

NB nI I

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

2oB N AN

LI

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 = L / R is the time required for the current to reach 63.2% of its maximum value

0dI

IR Ldt

1 tI t eR

9

RL Circuit, Current-Time Graph

The equilibrium value of the current is /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,

tdIe

dt L

12

13

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)

I 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

2 dII I R LI

dt

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

BdU dILI

dt dt

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

2 221

2 2oo o

B BU n A A

n

2

2Bo

U Bu

V

18

19

20

21

Inductance Example – Coaxial Cable Calculate L and

energy for the cable The total flux is

Therefore, L is

The total energy is

ln2 2

bo o

B a

I I bBdA dr

r a

ln2

oB bL

I a

221

ln2 4

o I bU LI

a

22

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