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

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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

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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

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Forces due to magnetic fields

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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

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Charge and current element

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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

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Example

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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

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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

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Magnetic Torque

Magnetic field in the plane of the loop

Rotation in clockwise direction

Total net force ??

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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

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Operation of DC Motor

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DC Motor (Commutator, brushes)

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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.