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Department of Semiconductor Systems Engineering SoYoung Kim
Engineering Electromagnetics- 1 Lecture 20: Magnetic Energy, Magnetic Forces
SoYoung Kim
ksyoung@skku.edu
Department of Semiconductor Systems Engineering
College of Information and Communication Engineering
Sungkyunkwan University
Department of Semiconductor Systems Engineering SoYoung Kim
Outline
8.9
8.1
8.3
Department of Semiconductor Systems Engineering SoYoung Kim
Magnetic energy
To assemble a group of charges Electric Energy
To send currents into conducting loop Magnetic Energy
An amount of energy to build up the current in inductor
Stored as magnetic energy
Single inductor L에 전류 I가 흐를때 저장되는 Magnetic Energy
P=vi
Department of Semiconductor Systems Engineering SoYoung Kim
0 to I1 0 to I2
C1 C2
1) C1에 저장되는 Energy (I2=0)
2) Induced EMF due to i2(i1 =I1) on C1 In order to keep i1 at I1 to overcome EMF
3) C2에 저장되는 Energy
Total amount of work done in raising the currents in C1 & C2
Magnetic energy
Department of Semiconductor Systems Engineering SoYoung Kim
Magnetic Energy
nSId0
S
sB
SnI
L2
0
Find I
Per unit length
2
2
1LIW
m
2
0
2
0)(
2
1
n
BSlnW
m
vBSlBWm
2
0
2
02
1
2
1
vdB
W
V
m
2
2
1
Department of Semiconductor Systems Engineering SoYoung Kim
Magnetic Energy
Total volume is assumed to be composed of differential volume below
2 2 2
2
0
1 1 1
2 2 2
1 1lim
2 2
1
2
m
m
mv
m m
H x zL
I I
I H y
W L I H x y z H v
Ww H B H
v
W w dv dv
B H
Department of Semiconductor Systems Engineering SoYoung Kim
Forces due to magnetic fields
Department of Semiconductor Systems Engineering SoYoung Kim
Force on a Charged Particle
Force on a charged particle due to electric field
Force on a charged particle due to magnetic field
Applies to only moving particles
The direction is perpendicular to the magnetic field direction
No work is performed on the charge
Total force on a charged particle due to fields
Lorentz force equation
eQF E
mQ F u B
( )Q F E u B
or ( )d
m Qdt
u
F E u B
Department of Semiconductor Systems Engineering SoYoung Kim
Force on a Current Element
Relation between a line current element and fractional charge
Magnetic force on a line current element
Magnetic force on a surface and volume current elements
v
I d dS dv
I d dQ
l K J
J u
l u
F u B
F l B
d dQ
d I d
S v
dS dv F K B F J B
Department of Semiconductor Systems Engineering SoYoung Kim
Charge and current element
Department of Semiconductor Systems Engineering SoYoung Kim
Force between Two Current Elements
Force on current element #1 due to B field created by current element #2
21
21
21
1 2
1 1 1 2
2 2
2 2
21
1 1 2 2
1 2
21
1 21 2
1 2
21
( )
4
( )( )
4
( )
4
o R
o R
Ro
L L
d d I d d
I dd
R
I d I dd d
R
d dI d
R
F l B
l aB
l l aF
l l alF
B caused by I2 in C2 F on C1
Ampere’s law of force F1 = -F2
Newton’s third law holds 작용반작용의 법칙
Department of Semiconductor Systems Engineering SoYoung Kim
Example
Department of Semiconductor Systems Engineering SoYoung Kim
Ex. 8.2
Department of Semiconductor Systems Engineering SoYoung Kim
Ex. 8.2
(b) Magnetic force
or
(c) K.E and Location
(d) Particle trajectory
Thus the particle gyrates in an orbit about a
magnetic field line
(e) KE constant
the uniform magnetic field has
no
effect on the K.E. of the particle.
Department of Semiconductor Systems Engineering SoYoung Kim
Ex. 8.4
Department of Semiconductor Systems Engineering SoYoung Kim
Ex. 8.4
The total force Fe on the loop is the
sum of F1, F2, F3, and F4
an attractive force trying to draw the loop toward the wire
Department of Semiconductor Systems Engineering SoYoung Kim
8.3 Torque
Department of Semiconductor Systems Engineering SoYoung Kim
Torque
F : Applied force
d : momentum arm
Distance vector measured from a point in the rotation axis to the point of application of F
Unit is same as work or energy but does not represent work or energy
Department of Semiconductor Systems Engineering SoYoung Kim
Magnetic Torque
Magnetic field in the plane of the loop
Rotation in clockwise direction
Total net force ??
Department of Semiconductor Systems Engineering SoYoung Kim
Magnetic Torque
Magnetic field perpendicular to the axis of the loop
Torque is max when B is parallel to the plane of the loop Torque is min when B is perpendicular to the plane of the loop
Department of Semiconductor Systems Engineering SoYoung Kim
Operation of DC Motor
Department of Semiconductor Systems Engineering SoYoung Kim
DC Motor (Commutator, brushes)
Department of Semiconductor Systems Engineering SoYoung Kim
Magnetic Torque and Moment
the force on a current loop in a uniform magnetic field
=0 net force is zero if B is uniform
Department of Semiconductor Systems Engineering SoYoung Kim
Magnetic Torque and Moment
Although this expression was obtained by using a rectangular loop, it is
generally applicable in determining the torque on a planar loop of any arbitrary
shape.
the torque is in the direction of the axis of rotation
It is directed with the aim of reducing a so that m and B are in the same
direction.
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