PART 2

PART 1 A. Electromagnetic Induction Phenomenon

B. Self- & Mutual-Induction Action

C. Electromotive Force due to Conductor Motion

D. Magnetic Energy of the Current Circuit System

E. Connection of the Inductance & Coupling Coefficient

F. Mutual Inductance between the Two Circuits

G. Examples of Inductance Calculation

H. Energy Stored in the Coil

I. Work by Electromotive Force

J. Eddy Current and Skin Effect

Ch.9. Electromagnetic Induction

yjshin@chosun.ac.kr www.chosun.ac.kr/~yjshin

A. Electromagnetic Induction Phenomenon

B. Self- & Mutual-Induction Action

C. Electromotive Force due to Conductor Motion

D. Magnetic Energy of the Current Circuit System

E. Connection of the Inductance & Coupling Coefficient

Yong-Jin Shin, Professor of Physics, Chosun University

PART 1

Ch.9. Electromagnetic Induction

9.A. Electromagnetic Induction

Phenomenon

Faraday’s Discovery

◈ Discovery of electromagnetic induction

• 1820 : Oersted

Demonstrate the impact of a compass by current

• 1831 : Faraday

Assumption : If the magnetic field can be generated by current,

magnetic field must be able to generate an electric field - Conviction

Goals : Current generated by the magnetic field

Conclusions : Changing current induces a current in the other conductor

◈ Induction Experiments

Faraday’s Discovery

Demonstrating the phenomenon of induced current

(a) Stationary magnet

(b) Moving the magnet toward or away from the coil

(c) Moving a second, current-carrying coil toward or away from the coil

(d) Varying the current in the second coil (by closing or opening a switch)

(a) It does Not induce a current in a coil.

(b)~(d) All these actions Do induce a current in the coil

Electromagnetic Induction

◈ Electromagnetic induction phenomenon

• Varying current (steady state current does not) induces a current in the other conductor.

• In other words, varying magnetic field (constant magnetic field does not) creates a current.

◈ Induced current

• Current generated by the electromagnetic induction phenomenon

◈ Induced electro-motive force (EMF)

• Corresponding emf required to cause induced current.

• Induced emf is proportional to the rate of change of magnetic flux

through the coil.

dt

demf

dt

dNemf

“Faraday’s law of

induction”

◈ Direction of induced EMF

Electromagnetic Induction

The magnetic flux is

becoming (a) more positive,

(b) less positive, (c) more

negative, and (d) less

negative.

Therefore flux is increasing

in (a) and (d), and

decreasing in (b) and (c).

In (a) and (d) the emfs are

negative (they are opposite

to the direction of the

curled fingers)

In (b) and (c) the emfs are

positive (in the same

direction as the curled

fingers)

thumb

curled fingers

Lenz’s Law

Convenient method for determining the direction of an induced current or emf.

The direction of any magnetic induction effect is such as to oppose the cause of the effect.

dt

d Induced emf by Faraday’s law of induction

C

dlE

emf by closed-circuit C

S

danB ˆ

Magnetic flux () through the cross-

sectional area S to create closed-circuit.

SSCdan

dt

BddanB

dt

d

dt

ddlE ˆˆ

By the above 3 formula

dandt

BddanEdanEdlE

SSSCˆˆˆ

By

Stoke’s theorem

t

BE

Thus, differential form of Faraday’s law

Lenz’s Law

◈ Direction of induced current Direction of induced currents as a bar magnet moves along the axis of a conducting loop. If the bar magnet is stationary, there is no induced current

In (a) and (d), the induced magnetic field is upward to oppose the flux change. To produce this induced field, the induced current must be counterclockwise as seen from the loop.

In (b) and (c), the induced magnetic field is downward to oppose the flux change. To produce this induced field, the induced current must be clockwise.

9.B. Self- & Mutual-Induction

Action

Self Induction

◈ Self Induction

• We consider only a single isolated circuit.

• When a current is present in a circuit, it sets up a magnetic field that

causes a magnetic flux through the same circuit; this flux changes when

the current changes.

• Thus any circuit that carries a varying current has an emf induced in it

by the variation in its own magnetic field.

• Such an emf is called “self-induced emf”.

• By Lenz’s law, a self-induced emf always opposes the change in the

current that caused the emf and so tends to make it more difficult for

variations in current to occurs

• Such these properties is called “self induction”.

Self-induced

emf

Self

Induction (Faraday’s Law)

Self-induced

emf

Self Induction

◈ Self Inductance (L)

i

NL

A

mT

A

WbH

2

111

The minus sign(−) is a reflection of Lenz’s law

• The current i in the circuit causes a magnetic

field in the coil and hence a flux through the

coil.

• If the current i in the coil is changing, the

changing flux through the coil induces an efm

in the coil.

• From Faraday’s law for a coil with N turns,

the self-induced emf is dt

dN

dt

diL

• Self-inductance of circuit is the magnitude of the

self-induced emf per unit rate of change of current.

• Thus, self inductance (coefficient of self induction) is

• Long solenoid with cross-sectional area A, closely wound with N turns

of wire ….

where, N = total number of turns

= magnetic flux

n = number of turns per unit length

l = total length of solenoid

niBwithBAnlN 0))((

• Calculating self inductance ?

LiN

lAn

i

Aninl

i

BAnl

i

NL 2

00

• What is the self inductance in the central part of the long solenoid ?

Anl

L 2

0

Self Inductance (Solenoid) G. Examples of Inductance Calculation

• Toroidal solenoid with cross-sectional area A and mean radius r is closely

wound with N turns of wire.

• Calculating self inductance ?

)(BANN l

Ni

r

NiBwith 00

2

LiN

i

BAN

i

NL

)(

i

Al

NiN

0

l

AN 2

0

Self Inductance (Toroid) G. Examples of Inductance Calculation

Self Inductance (Toroidal Coil)

• Calculating self inductance ?

)(BANN r

NiBwith

2

0

b

ahdr

r

NiBdA

2

0

b

a r

drNih

2

0

a

bNihln

2

0

LiN

a

bhN

a

bNhN

i

NL ln

2ln

2

2

00

• Toroidal Coil with rectangular cross-sectional area A and average length

of circumference l is closely wound with N turns of wire.

G. Examples of Inductance Calculation

Mutual Induction

A current i1 in coil 1 gives rise to a

magnetic flux through coil 2.

If the current in coil 1 is changing, the

changing flux through coil 2 induces an

emf in coil 2.

This can be explained by the mutual

inductance.

A) Current flowing i1 in coil 1 produces magnetic flux through coil 2.

We denote the magnetic flux through each turn of coil 2 as 21

121212 iMN 21M : Mutual Inductance (constant)

It depends only on the geometry of the two coils

(size, shape, number of turns, orientation of

each coil and the separation between the coils) Induced emf in coil 2 ….

dt

diM

dt

dN 1

2121

22

Negative sign (-) means Lentz’s law

B) Current flowing i2 in coil 2 produces magnetic flux through coil 1.

Magnetic flux through each turn of coil 1 as …. 12 212121 iMN

Induced emf in coil 1 ……

dt

diM

dt

dN 2

1212

11

C) M21 = M12 = M (Reciprocity Theorem)

Induced emf in coil 1 and coil 2 → mutually induced emfs

dt

diM 1

2 dt

diM 2

1 and

2

121

1

212

i

N

i

NM

Mutual Inductance (coefficient of mutual inductance)

2)(

A

Js

A

sV

A

WbhenryH

단위 :

Mutual Induction