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TRANSCRIPT
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
[email protected] 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