(1)2008, 1st Electromagnetic field 1 Syllabus (2)(3)Representation : A, A, A, A, L ~ h A j A i Ai ˆ+ j ˆ+ h ˆ = 방향의 성분 i Unit vector 2-2 Vector Addision and Subtraction [Text p.12] A B B A

2008, 1st Electromagnetic field 1 Syllabus

Representation : A, A, A, A, L

~

h A j A i

A_i ˆ+ _j ˆ+ _h ˆ

=

방향의 성분

i Unit vector

2-2 Vector Addision and Subtraction [Text p.12]

A B B

A+ = + : Commutative law

C B A C B

A+( + )=( + )+ : Associative law

2-3.1 Scalar (or dot, inner) product

θ cos AB B

A⋅ Δ ^θ _B

A

A B B A⋅ = ⋅

C A B A C B

A⋅( + )= ⋅ + ⋅

: Commutative law

: Distributive law

Chapter 2. Vector Analysis

Scalar Field (magnitude) : temperature, density, electric potential Vector Field (magnitude and direction) : velocity, force

Electric field intensity E, Electric displacement D Magnetic flux density B, Magnetic field intensity H

Field “A ftn which describes a physical quantity in space”

L , , , , A A A A

A

2-1 Introduction [Text p.11]

2-3 Products of Vectors [Text p.14]

2-3.2 Vector (or cross, outer) product

2-4 Orthogonal Coordinate Systems [Text p.20]

θ ˆ ABsin a B A× Δ _n

: unit vector perpendicular to

3 3

2 2

1 1

) , , (

) , , (

) , , (

g z y x U

g z y x U

g z y x U

=

=

=

each represents a family of surfaces in space

Intersection of Intersection of Intersection of

) , ,

(x_o y_o z_o P

aˆn ^{A and B}

formed according to right-hand rule

B

A aˆn

consider

surfaces surfaces surfaces

curve curve curve

3

2, U

U

1

3, U

U

2

1, U

U

U1

U2

U3

Three surfaces intersect at one pt

x

z

y U1

U3

U2

P ˆ_u3

a ˆ_u2

a ˆ_u1

a

curve

curve

curve

3 2 1, u , u P u

3 2

1, ˆ , ˆ

ˆ_u a_u a_u

a : Units vectors issuing from target to curve

( or orthogonal to q₁, q₂, q₃ surfaces ) When three surfaces intersect one another orthogonally

orthogonal coordinate sys.

3 2

1 ˆ ˆ

ˆ_u a_u a_u

a × =

, ˆ 0 ˆ_u1⋅ a_u2=

a ˆ ˆ 0, L

3

2⋅ _u =

u a

a

1 ˆ ˆ_u1⋅ a_u1= a

Then, the Length element

[ ]

[ 3 3 ²]¹²

2 2 2 2 1 1

2 1 2 3 2 2 2 1

) ( ) ( ) (

) ( ) ( ) (

du h du

h du

h

dl dl

dl dl

+ +

=

+ +

=

Volume element

3 2 1 3 2

1h h du du du

h dv=

Let length elements are related todl_i Coordinate variables by

i i

i h du

dl =

Where : metric coefficienth_i

ui

Cartesian Cylindrical Spherical

h1

h2

h3

1 1 1

1 1

1

ρ _r

θ sin r )

, ,

(x y z (ρ,φ,z) (r,θ,φ)

2-4.1 Cartesian coord

) , , ( ) , ,

(u₁ u₂ u₃ = x y z

1 ,

1 ,

1 _{2 3}

1 = h = h =

h

dz dy dx dv=

) , , (x y z P

x

y z

2-4.2 Cylindrical coord

) , , ( ) , ,

(u₁ u₂ u₃ = r φ z

1 ,

,

1 _{2 3}

1= h =r h =

h

φ r z

z

x

y

• P

φ θ θdRd d R

dv= ²sin 2-4.3 Spherical coord

) , , ( ) , ,

(u₁ u₂ u₃ = R θ φ

θ sin ,

,

1 _{2 3}

1 h R h R

h = = =

P

x

y z

R θ

φ

•

dz rdrd dv= φ

Invariance

Symbolically P^' = RP where

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

−

=

=

=

1 0 0

0 0 θ θ

θ θ

cos sin

sin cos

) , , ( ),

, , ( ^{' ' '}

'

R

z y x P z y x P

Vectors

and

' ' ' ' ' ' '

ˆ ˆ ˆ

ˆ ˆ ˆ

k z j y i x P

k z j y i x P

+ +

=

+ +

=

' ' ' ' ' ' '

ˆ ˆ ˆ

ˆ ˆ ˆ

k w j v i u Q

k w j v i u Q

+ +

=

+ +

=

Scalar product of and

Q P

zw yv

xu

zw v

u y

x

v u

y x

w z v y u x Q P

zw yv xu Q P

⋅

=

+ +

+ +

=

+ +

− +

− +

+ +

=

+ +

=

⋅

+ +

=

⋅

) sin (cos

) sin (cos

) cos sin

)(

cos sin

(

) sin cos

)(

sin cos

(

2 2

2 2

' ' ' ' ' ' ' '

θ θ

θ θ

θ θ

θ θ

θ θ

θ θ

P Q

Q P⋅

∴ are independent of coordinate sys

: invariant (A quantity that is independent of coordinate system) ,

), ( , ,

, flux P

dt p Q d

P× ∇ Φ ∇⋅ are all invariant

z z

y x

y

y x

x

=

+

−

=

+

=

' ' '

cos sin

sin cos

θ θ

θ θ

Rotation of coordinate system by an angle about Z-axis

coordinates and of θ

) , ,

(x y z (x^', y^', z^') P

X X'

Y

Y'

θ P

2-5 Integrals Containing Vector Functions[Text p.37]

Line integral

[ ]

∫

∫

∫

∫

∫

+

+

=

+ +

=

c z

c y c

x c

z y

x c

dz z y x V a

dy z y x V a dx z y x V a

dz a dy a dx a z y x V l

Vd

) , , ˆ (

) , , ˆ (

) , , ˆ (

ˆ ˆ

) ˆ , , (

Surface integral A dS

s

∫

⋅

Volume integral Fdv

v

∫

Closed line or surface

∫

c

or

∫

s

(Ex) A = Ezˆ

1.

2.

1 : x² + y² + z² =

S 로 주어진 반구

2 1

2 + =

= x y

S ^{로 주어진 원판}

S d A

s

∫

⋅

find x

y z

2-6 Gradient of a Scalar Field [Text p.42]

Thus

3 2

1

3 2

1 ˆ ˆ

ˆ l

a V l a V l a V

V _{u u u}

∂ + ∂

∂ + ∂

∂

= ∂

∇

gradient의 성질

a. magnitude is the maximum rate of change with distance b. direction is that of the maximum rate of change c. it points towards larger values of ftn.

grad

dn a dV V

VΔ∇ Δ ˆ_n ∇ : del

( 1 2 3)

3 2

1

3 3 2 2 1 1

3 2

1 3

2

1 ˆ ˆ ˆ ˆ ˆ

ˆ a dl a dl a dl

l a V l a V l a V

l dl dl V

l dl V l V

l d V dV

u u

u u

u

u ⎟⎟⎠⋅ + +

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂

= ∂

∂ + ∂

∂ + ∂

∂

= ∂

⋅

∇

=

where aˆ_n : direction of maximum rate of increase

Directional derivative

l l

n a V a

dn a dV dl dn dn dV dl

dV = = ˆ ⋅ˆ =∇ ⋅ ˆ

l d

3 3 2

2 1

1 ^{2 3}

1 ˆ ˆ

ˆ h u

a V u h a V u h a V

V _{u u u}

∂ + ∂

∂ + ∂

∂

= ∂

∇

2-7 Divergence of a Vector Field [Text p.46]

v S d A A

div v Δ

Δ ∫ ⋅

→ Δlim0

) , ,

(x_o y_o z_o P

x

y z

•

Δy

Δx Δz

1.

(net outward flux / vol)

z x y

x z Ax y x A

z y z x y

x A

q z y A S d A

o o

o y z

x o

o o x

o o o

x

x

Δ

⎪⎭Δ

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ Δ +

⎥⎦⎤

∂ + ∂

=

Δ Δ Δ

+

=

Δ Δ

⋅

=

⋅

2 L )

, , (

) , 2 , (

)ˆ (

) , , (

for surface

for surface

z l y

x z Ax y x A S

d A

o o o y z x o

o o

x Δ Δ

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ −Δ +

⎥⎦⎤

∂ +∂

−

=

⋅ ) L

( 2 )

, , (

) , , (

Flux of over a surface A S S

d A

s

⋅

=

Φ ∫ 미소 면적 성분, 수직 방향

2.

1.

2.

3.

4.

5.

6.

[ ]

^{A dS}

S d

A⋅ =

∫ ∫ ∫ ∫ ∫ ∫

+ + + + + ⋅

∫

1. 2. 3. 4. 5. 6.

[ ]

Δ Δ Δ +LL

∂

= ∂

⋅

∫ ∫+ ^{A dS Ax}_x ^{x y z}

같은 방법으로,

[ ]

Δ Δ Δ +LL

∂

= ∂

⋅

∫ ∫+ ^{A dS Ay}_y ^{x y z} and

[ ]

Δ Δ Δ +LL

∂

= ∂

⋅

∫ ∫+ ^{A dS Az}_z ^{x y z}

and

L +L Δ Δ

⎟⎟Δ

⎠

⎜⎜ ⎞

⎝

⎛

∂ +∂

∂ +∂

∂

= ∂

∫ ^A⋅^{dS Ax}_x ^Ay_y ^Az_z ^{x y z} and

z Az y

Ay x

A Ax div

A ∂

+ ∂

∂ + ∂

∂

= ∂

≡

⋅

∇

For general orthogonal curvilinear coord

⎥⎦

⎢ ⎤

⎣

⎡

∂ + ∂

∂ + ∂

∂

= ∂

⋅

∇ 1 ( ) ( ) ( )

3 2 1 3 2

1 3 2 1

3 2 1 3 2 1

A h u h A

h u h A

h u h h h A h

1. 2.

3. 4.

5. 6.

S d A v

A

sj

N j j j N

j ∇⋅ Δ = Σ ⋅

Σ=1( ) =1 ∫

S d A dv

A

s v

⋅

=

⋅

∇ ∫

∫

Nonzero divergence source or sink of flow For an arbitrary vol.

Divergence Theorem.

⋅ A

∇ : measure of strength of flow source 2-8 Divergence Theorem [Text p.50]

2-9 Curl of a vector field [Text p.54]

(Ex) 만일

Circulation of around contourA c A d l

C

⋅ Δ

∫

F

A = (force)

Circulation of ^A=

∫

^F ⋅^{d l} =

C

work curl

max

0 1 ˆ

lim ⎥

⎦

⎢ ⎤

⎣

⎡ ⋅

Δ Δ

×

∇

=

∫

→

Δ a A d l

S A A

C S n

aˆn

l d ΔS

aˆn : normal direction of area right hand rule

l d y A

x

l d y A

A x

y x

y z x

Δ ⋅

= Δ

Δ ⋅

= Δ

×

∇

∫

∫

→ Δ Δ

→ Δ Δ

lim 1 lim 1 )

(

0 0

along contour

ax

x l

d =Δ ˆ

Δy

Δx

x y

•

) , ,

(x_o y_o z_o P

1. 2.

3.

4.

1. 2. 3. 4.

] ^{y x}

y z A

y x A

x y z

y x A l d A

o o o y z x x o

o o x

o o

o x

⎥Δ

⎦

⎢ ⎤

⎣

⎡ ⋅ −Δ +

∂ + ∂

=

Δ Δ

−

=

∫

⋅

L 2 L

) ) (

, , (

) 2 , ,

(

, ,

1. 1.

along contour2.

ay

y l

d = Δ ˆ

x y x z Ay

y x A

y z x y

x A l d A

o o o y

o o o

y

⎥⎦Δ

⎢⎣ ⎤

⎡ ⋅Δ +

∂ + ∂

=

Δ Δ +

=

∫ ⋅

L 2 L

) , , (

) , 2 ,

(

along contour3.

ax

x l

d = −Δ ˆ 2.

)

2y ( x

y A Ax

l d

A _x ⎥ −Δ

⎦

⎢ ⎤

⎣

⎡ Δ +

∂ ⋅ + ∂

=

∫

⋅ ^LL

3.

along contour4.

) 2 (

)

( x y

x A Ay

l d

A _y −Δ

⎥⎦⎤

⎢⎣⎡ ⋅ −Δ +

∂ + ∂

=

∫

⋅ ^LL

ay

y l

d = − Δ ˆ

4.

y x x

y A y x

l A d

A ^{x y} Δ Δ

∂ + ∂ Δ

∂ Δ

− ∂

=

⋅

∴

∫

y A x

A _z A^{y x}

∂

− ∂

∂

= ∂

×

∇ )

(

같은 방법으로,

z y z

y z x

x

z y a

y A x

a A x A z

a A z A y

A A ˆ ˆ ⎟⎟⎠ˆ

⎜⎜ ⎞

⎝

⎛

∂

− ∂

∂ + ∂

⎟⎠

⎜ ⎞

⎝

⎛

∂

− ∂

∂ + ∂

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂

− ∂

∂

= ∂

×

∇

In general orthogonal curvilinear coord system

= 0

×

∇ A

z y

x

z y

x

A A

A

z y

x

a a

a

∂

∂

∂

∂

∂

= ∂

ˆ ˆ

ˆ

3 3 2

2 1

1

3 2

1

3 2

1

3 2 1

3 2

1 ˆ ˆ

ˆ

1

A h A

h A

h

u u

u

a h a

h a

h

h h A h

u u

u

∂

∂

∂

∂

∂

= ∂

×

∇

A is a conservative field

2-10 Stoke’s theorem [Text p.58]

From the definition of curl, i.e.

⎥⎦

⎢ ⎤

⎣

⎡ ⋅

= Δ

×

∇ Δ →

∫

C

S an A d l

A 1S ˆ

lim0

For surrounded by ΔS

l d A a

S A

C

n = ⋅

Δ

⋅

×

∇ ^{) ˆ}

∫

(

C

For an arbitrary surface , subdivide it into many S N and add up all the differential areas

S d A S

j A

s j N

j

S_j→ Σ= ∇× ⋅Δ =

∫

∇× ⋅

Δlim ( )

1 0

l d A l

d A

c c

N j S

j j

⋅

=

⋅ Σ

= Δlim→ = (

∫

)

∫

1 0

l d A S

d A

C S

⋅

=

⋅

×

∇ ∫

∫^{( )}

∴ Stoke’s theorem

C

Sj

Δ

S

2-11 Two Null Identities [Text p.61]

( ^∇ ) ^{= 0}

×

∇ V

2-11.1 Identity I

pf. By Stoke’s theorem

[

∇ ×

( )

∇

]

⋅ =

∫ ( )

∇ ⋅ =

∫

= 0

∫

^{V da V d l dV}

s

V E

E = = − ∇

×

∇

← If 0,

l d E l

d E

b

a

⋅

→

=

⋅

∫

∫

⁰

: depends only on end pts a and b conservative field

a

•b

•

1.

2.

2-11.2 Identity II

( ^{∇ ×} ) ^{= 0}

⋅

∇ A

pf. By Divergence theorem

( ) ( )

( ) ( )

0

2 1

2 1

=

⋅ +

⋅

=

⋅

×

∇ +

⋅

×

∇

=

⋅

×

∇

=

×

∇

⋅

∇

∫

∫

∫

∫

∫

∫

l d A l

d A

a d A a

d A

a d A d

A

c c

S S

s v

τ

C1

C2

S2

S1

V

←

^If

∇ ⋅ B = 0 → B = ∇ × A

magnetic flux density

vector potential

Chap 3. Static Electric Fields

Field : Spatial distribution of scalar and vector quantity Electric field and magnetic field.

3-1 Introduction

[Text p. 72]

Electrostatics : 시간에 따라 변하지 않는 전하분포 혹은 전장과 관계된 전기 현상

ˆ

12

4 1

2 12

2 12 1

R o

R a q F q

= πε

Coulomb’s law (1785)

(Newton)

q

¹

, q

²

R

12

: coulomb : m

where

ε

_o

= 8 . 84 × 10

⁻¹²

( F / m )

Permittivity of free space

Force exerted on by

q

₂

q

₁

q

2

ˆ

_R12

a

Electric field intensity

3-2 Fundamental Postulates of Electrostatics in Free Space

[Text p. 74]

Integral form of two postulates

) /

(

lim

₀

V m q

E F

q→

=

Postulate 1.

Postulate 2.

o

E ε

= ρ

⋅

∇

ρ : charge density

( C / m

³

)

= 0

×

∇ E

Postulate 1

∫

∫

∇ ⋅ = =

V

o V

o

dv Q dv

E ρ ε

ε

1

Q : total charge in ^V

o S

S Q d

E ^{⋅ =} ε

∫

: Gauss law

since

0

2 1

=

⋅ +

⋅

=

⋅ _{∫ ∫}

∫ E d l E d l E d l

C C

Postulate 2.

l d E

l d E

l d E

P P

P P P

P

⋅

=

⋅

−

=

⋅

∫

∫

∫

2 1

1 2 2

1

= 0

⋅

=

⋅

×

∇ _∫

∫ E d S E d l

C S

•

• P

1

P

2

C

1

C

2

1을 따라

C C

₂

C

2

line integral of is independent of path depends only on end points

conservation of energy

E

를 따라

를 따라

3-3 Coulomb’s law

[Text p.77]

Electric field due to point charge applying Gauss law

(Ex 3-2) Electric field inside of a spherical shell with a total charge Q

o R

S R

R

S R R

S

q

R E

dS E

dS a

a E S

d E

ε π

/ 4

) ˆ

( ) ˆ (

2

=

=

=

⋅

=

⋅

∫

∫

∫

q • R

m V

R a E q

a

E

_R

o R

R

ˆ /

ˆ 4

₂

= πε

=

• P S1

1. By Gauss law

2. Field at due to and

P S

1

S

₂

4 ₂ ² 0

2 2 1

1 ⎟ ⎟ =

⎠

⎞

⎜ ⎜

⎝

⎛ −

=

r dS r

dE dS

o

πε s

ρ

dS r

Solid angle

r

2

d Ω = dS

Tilted surface

α

2

cos

r d Ω = dS

α

3-3.1 Electric field due to a system of discrete charges [Text p. 82]

( )

k k

n k k o

R

q

R R

R

E R

1 3

1

) 1

(

4

1

− Σ −

=

₌

πε

Principle of superposition

Electric field : linear ftn of

a

_R

R q ˆ

• •

•

• •

• R•

R2

R1

q1

q2

0 Continuous distribution

v R d

a E

v R

o

= _∫ ′

′

ˆ

2

4

1 ρ

πε

s R d

a

E

^S

S R

o

= _∫ ′

′

ˆ

2

4

1 ρ

πε

l R d

a

E

^l

L R

o

= _∫ ′

′

ˆ

2

4

1 ρ

πε

: Volume charge : Surface

: Line

• Electric dipole : sys of charges consists of a pair of equal and opposite charges

+ q

^and

− q

separated by a small distance

d << R

Electric field at

P

같은 방법으로,

Z P

R

R

−

R

+

θ + q

− q 0

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ + ⋅

−

⎟ ⎠

⎜ ⎞

⎝

⎛ − ⋅ +

= −

−

−

2 3

2 3 2 2

3

2 1 3

~ 2

4 2

2 2

R d R R

R d

d d R R

R d

R d R d

3

3 2

/

2 1 3

2 2

2 R

R d R R d

R d R d

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ − ⋅

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ +

= +

+

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ − + ⋅

= R

R d d R

R E q

o

2

3

3

1 4 πε

and

d q

P =

: Electric dipole moment

⎥ ⎥

⎥ ⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎣

⎡

+

− +

−

= −

3 3

2 2 2

2

4 d

R R d

R d R d E q

πε

o