PDF Field and Wave Electromagnetics - Seoul National University

Field and Wave Electromagnetics

Chapter 3 Static Electric Fields

Chapter 3. Static Electric Fields

3 1 Introduction 3-1 Introduction

Coulomb’s law Experimental law Coulomb’s law Experimental law

ˆ q q1 2

F k

12

12 1 22

12

ˆ_R q q

F a k

= R

3 2 Fundamental Postulates of Electrostatics in Free Space 3.2 Fundamental Postulates of Electrostatics in Free Space

Force : F (N) Force :

El i fi ld i i (전계 강도) (N)

F

Electric field intensity (전계 강도)

F

lim0 [ V m ] or [ N C]

q

E F

→ q

=

why? - Not to disturb the charge distribution why? Not to disturb the charge distribution

of the source

3 2 Fundamental Postulates of Electrostatics in Free Space Two fundamental postulates of electrostatics

3.2 Fundamental Postulates of Electrostatics in Free Space

Two fundamental postulates of electrostatics

E ρ

⎧ ∇ = ε

⎪⎨ i ―① (MKS) ^{note ∇iE = 4πρ (cgs)}

0

0 E

⎨ ε

⎪ ∇× = →

⎩ irrotational ―②

0

1

v Edv v dv

Q ε ρ

→

∫

∇ =

∫

∫

i

①

0

S

E d s Q

=

∫

i = ε Gauss's law 0

E d l

→

∫

ⁱ =

② 0

( ) 0

c

S

E d l

E ds E d l

→ =

∇× = =

∫

∫ ∫

i

∵ i i

②

Kirchhoff’s voltage law

3 2 Fundamental Postulates of Electrostatics in Free Space Conservative field

3.2 Fundamental Postulates of Electrostatics in Free Space Conservative field

1 2

0

C E d l + C E d l =

∫

₁ ⁱ

∫

₂ ⁱ

2 1

1 2

1 2

Along Along

P P

P P

C C

E d l = − E d l

∫

ⁱ

∫

ⁱ

1 2

2 1

2

g g

Along

^P

P C

E d l

=

∫

ⁱ

g 2

Why?

E d l E

∫

Why?

(Usually depends on integral path) cf) is irrotational!!

i

∵E

cf) ∵ is irrotational!!

3 3 Coulomb’s Law 3-3 Coulomb s Law

Coulomb’s law Coulomb’s law

ˆ ˆ

( _R)

S S

E d s = R E R ds = q

∫

ⁱ

∫

ⁱ

0

2 0

( )

4

S S R

R

E q

R

ε

= πε

∴

∫ ∫

0

2 0

4

ˆ ˆ [V m ] or [ N C]

R 4

R E R E R q

R πε

= = πε

General Expression

2 3

0 0

( )

ˆ

4 4

p qp

q q R R

E a

R R R R

πε πε

− ′

= =

′ ′

− −

1 2 2 0

ˆ [N ]

4 F R q q

πε R

= cf)

3 3 Coulomb’s Law

Charged Shell

3-3 Coulomb s Law

Charged Shell

1 2

s ds ds

dE ρ ^{⎛ ⎞}

= ⎜ 2 − 2 ⎟

0 1 2

2 2

1 2

4

cos

dE r r

d r d r

πε

α

= ⎜ ⎟

⎝ ⎠

Ω ⋅ = = Ω ⋅ solid angle

1 2

cos 0

ds ds

dE

α

= =

∴ =

solid angle

Discrete Charges Discrete Charges

3 1

( )

1 [V m]

4

n

k k

k

q R R E

R R πε

− ′

=

∑

′

0 1

4 _k

R Rk

πε _{= − ′}

3 3 1 Electric Field due to a System of Discrete Charges 3-3.1 Electric Field due to a System of Discrete Charges

Electric dipolep

2 2

d d

R R

q

⎧ ⎫

⎪ − + ⎪

⎪ ⎪

3 3

0

2 2

4

2 2

E q

d d

R R

πε

⎪ ⎪

= ⎨ − ⎬

⎪ − + ⎪

⎪ ⎪

⎩ ⎭ ₃

3 3 3

2

2 2

2

3 cos

2 2 2 4 ,

Rd

d d d d

R R R R R d R d

θ

− ⎡⎛ ⎞ ⎛ ⎞⎤− ⎡ ⎤−

− = ⎢⎜ − ⎟ ⎜⋅ − ⎟⎥ = ⎢ − ⋅ + ⎥

⎢⎝ ⎠ ⎝ ⎠⎥ ⎣ ⎦

⎣ ⎦

⎡ ⎤

3 2

3 3

2 2

1 1 3 (binomial expansion)

2

R d R d

R R

R R

−

− −

⎡ ⎤

⎡ ⋅ ⎤ ⎢ ⋅ ⎥

≅ ⎢ − ⎥ ≅ ⎢ + ⎥

⎣ ⎦ ⎢ ⎥

⎣ ⎦

2 3

( 1) ( 1)( 2)

(1 ) 1

2! 3!

n n n n n n

x nx x x

⎣ ⎦

− − −

± = ± ± ±

note

3 3 q R d

d ⁻ R d

±

⎡ ⎤ ⎡ ⎤

0

2 3 2

3 3

3

2 1 2 4

q R d

E R d

R R

d R d

R R

R πε

− ⎡ ⋅ ⎤

+ ≅ − ∴ ≅ ⎡⎢ ⋅ ⋅ − ⎤⎥

⎢ ⎣

⎣ ⎥ ⎦

⎦

3 3 1 Electric Field due to a System of Discrete Charges 3-3.1 Electric Field due to a System of Discrete Charges

3 2

( )

1 3 , ˆ cos ˆ sin ˆ

4

p qd electric dipole moment vector

E R p R p z R

R R θ θ θ

=

⎡ ⎤

= ⎢ ⋅ − ⎥ = −

⎣ ⎦

i Define

then,

( )

( )

3 2

4 0

cos

ˆ

ˆ i

R R

R p Rp R πε

θ

θ θ θ

⎢ ⎥

⎣ ⎦

=

i z

zˆ

θ ^Rˆ

( )

( )

3 0

cos sin

ˆ ˆ

2 cos sin [V/

4 p p R

E p R

R

θ θ θ

θ θ θ πε

= −

= + m] θ

θˆ

y

x

3 3 2 Electric field due to a continuous distribution of charge 3-3.2 Electric field due to a continuous distribution of charge

Continuous distribution Continuous distribution

2 ' 2

0 0

' 1

ˆ ˆ ', where ˆ

4 4 ^v

dv R

d E R E R dv R

R R R

ρ ρ

πε πε

= ⎯⎯→ =

∫

=

0 0

' 3 0

4 4

1 ' [V/m]

4 ^v

R R R

E R dv

R

πε πε

πε ρ

=

∴

∫

) Surface & line charge density case cf

0

' 2

0

) Surface & line charge density case

1 ˆ

' [V/m]

4

s S

cf

E R ds

R ρ

= πε

∫

' 2

0

1 ˆ

' [V/m]

4

l

E L R dl

R ρ

= πε

∫

Ex 3 4 Infinitely long line charge Ex 3-4 Infinitely long line charge

Determine the electric field intensity of an infinitely long,

ˆ ˆ

sol) ', _l ' _l '

l

R r r z z ρ dl ρ dz

ρ

= − =

Determine the electric field intensity of an infinitely long, straight, line charge of a uniform density in air

3

0 0

ˆ ˆ

' ' '

4 4

l dz R l dz r r z z

d E R

ρ ρ

πε πε

= = − ₃

2 2 2

ˆ ˆ

( ' )

'

r z

r dE zdE r z

d

= +

+

3

2 2 2

0

where '

4 ( ' )

' '

l r

dE r dz

r z z dz ρ

πε ρ

=

+

3

2 2 2

0

( ' )

'

l z

l

dE z dz

r z dE r dz

ρ πε

ρ

= −

4 +

⎧⎪ ³

2 2 2

0

3

4 ( ' )

at ',

' '

l r

l

dE

r z z z

dE z dz

ρ πε

ρ

⎪ =

⎪ +

= ⎪⎨⎪ = −

⎪ ^' _{2 2} ³

( ' )2 z z z

dE

r z

= πε

⎪ 4 +

⎪⎩

Ex 3 4 Infinitely long line charge Ex 3-4 Infinitely long line charge

⎧

3

2 2 2

0

'

4 ( ' )

at ',

' '

l r

dE rdz

r z

z z

d ρ

πε

⎧ =

⎪⎪ +

= − ⎪⎨

⎪

' 3 '

2 2 2

0

' '

( ' )

'

l

z z z z z z

dE z dz dE

r z

r dz

ρ πε

ρ ρ

=− =

∞

⎪ = = −

⎪ 4 +

⎪⎩

∫ ³

2 2

0 2 0

ˆ ˆ ˆ [V/m]

( ' ) 2

_r ^l r dz ^l

E r E r r

r z r

ρ ρ

πε πε

∞

∴ = = −∞ =

4 ∫ +

HW. Integrate to have the above result cf) Symmetry →Gauss law

3 4 Gauss’s Law and Applications 3-4 Gauss s Law and Applications

C lind ical s mmet Cylindrical symmetry

( )

^Q

E ρ E dv

∇ⁱ = ⇒

∫

_'

( )

∇ⁱ =

0 0

By Divergence theorem

0

v

s

E ds Q

ε ε

←⎯⎯⎯⎯⎯⎯→ =ε

∫

∫

ⁱ

z 0 E =

ε0

2 2 ,

L lL

E ds = ^π E r d dzφ = πrLE = ρ

∫

ⁱ

∫ ∫

Er

0 0

0

2 ,

where, , 0

r r

S

r Z

E ds E r d dz rLE E rE E

φ π

ε

= =

∫ ∫ ∫

z 0 E =

0

2

l

Er

r ρ

∴ = πε

Ex 3 7 Spherical electron cloud Ex 3-7 Spherical electron cloud

for 0 R b

ρ = ρ ≤ ≤

⎧ _{0 , for 0} Spherical symmetry → i e E = r Eˆ

0 , for R b R b

ρ ρ

ρ

= − ≤ ≤

⎧⎨ = >

⎩

2

Spherical symmetry i.e.

i) 0 b

4

r

R R

E r E R

E ds E ds E πR

→ =

≤ ≤

⋅ = =

∫

S_i

∫

3

0 0

4

3

i

R S R

v v

Q dv dv R

π

ρ ρ ρ π

= = − = − ⋅

∫ ∫

∫ ∫

0 0

ˆ , 0

E R 3ρ R R b

= − ⋅ ε ≤ ≤

∴

3

ii) b

4 (fixed)

R

Q ρ π b

≥

= ₀

3 0

(fixed)

3 ˆ

3

Q b

E R b ρ

ρ ε

=

∴

−

−

= ₂ , R b

R ≥

3ε₀R

3 5 Electric Potential 3-5 Electric Potential

2

1

0 (Vector Identity)

^P [J/C or V] (Work must be done against the field)

P

E E V

W E dl

q

∇× = → =−∇

=−∫ ⋅

2 2 2 2

1 1 1 1

2 1 ˆ

^P [V] ^{P P} ( ) ( ) ^P

P P P P

q

V − =−V ∫ E dl⋅ ^∵−∫ E dl⋅ =−∫ −∇V ⁱ l dl =∫ dV Remember when we define gradient

ˆ ˆ

dV dV dn dV cos dV ˆ ( ) n l V l dl = dn i dl = dn α = dn i = ∇ i

( )

dV = V dl

∴ ∇ i

) Refernece point of potential: the zero potential point is usually at infinity cf ) Refernece point of potential: the zero potential point is usually at infinity.

cf

3 5 1 Electric potential due to a charge distribution 3-5.1 Electric potential due to a charge distribution

Point charge at origin potential at R w r t that at infinity Point charge at origin, potential at R w.r.t. that at infinity

2 0

ˆ 4 E R q

πε R

=

2

0 0

ˆ ( ˆ ) [V]

4 4

R q q

V R R dR

R R

πε πε

∞

⎛ ⎞

= − ⎜ =

∴ ⎠⎟

∫

⎝ ⁱ

V₂₁ : Potential difference between point P₁ and P₂

2 1

21 P P

0 2 1

1 1

4 V V V q

R R

πε

⎛ ⎞

= − = ⎜ − ⎟

⎝ ⎠

3 5 1 Electric potential due to a charge distribution 3-5.1 Electric potential due to a charge distribution

No work ˆ

_r ˆ dl R d F F r

φ φ

=

= No work i

0 F dl=

∴ i

1 [V]

n

qk

V

∑

^'

0 1

4 [V]

k

k k

V ⁼ πε

∑

₌ R ₋ R

Ex 3 8 An Electric dipole Ex 3-8 An Electric dipole

1 1

q ⎛ ⎞

0

1

1

1 1

4

1 cos 1 cos

2 2

V q

R R

d d

R R

R R

πε

θ θ

+ −

−

−

⎛ ⎞

= ⎜ − ⎟

⎝ ⎠

⎧ ⎛ − ⎞ ≅ ⎛ + ⎞

⎪ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠

⎪

1

1

2 2

,

1 cos 1 cos

2 2

R R

d R

d d

R R

R θ R θ

+

−

−

−

⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎪⎨

⎛ ⎞ ⎛ ⎞

⎪ ⎜ + ⎟ ≅ ⎜ − ⎟

⎪ ⎝ ⎠ ⎝ ⎠

⎩

2 2

0 0

cos 1 ˆ

, where : dipole moment vector

4 4

V qd p R p qd

R R

E

θ

πε πε

= = ⋅ =

= −∇^{V = −Rˆ ∂}_∂^V_R ^−θˆ _R^∂_∂^V_θ ⁼ ₄ ^p_R³³

( (

^{Rˆ 2cosθ θ+ ˆsinθ}

) )

0

1 2 3

1 1 2 2 3 3

4

ˆ ˆ ˆ

)

R R R

V V V

cf V u u u

h u h u h u

θ πε

∂ ∂

∂ ∂ ∂

∇ = + +

∂ ∂ ∂

2

Note

1. Equipotential line : cos

2 fi ld li i

R cV

E R

θ θ

= 2. E−field line : R = c_E sin2θ

Potential due to continuous charge

'

1 '

4 ^v

V dv

R ρ

= πε

∫

[V] : volume charge distribution

0

0 '

4

1 '

4

v

s

R

V ds

R πε

ρ

= πε

∫

∫

[V] : surface charge distribution

0 '

1 '

4

l

V L dl

R ρ

= πε

∫

[V] : line charge distribution

Ex 3 9 A uniformly charged disk Ex 3-9 A uniformly charged disk

Obtain a formula for the electric field intensity

2 2

2

1

y

sol) ' ' ' ' and '

^{s b} ' ' '

ds r dr d R z r

V ^π r dr d

φ

ρ φ

= = +

= ∫ ∫

( )

( )

0 0 1

2 2

0 2

1

2 2 2

' '

b s

z r

z r z

πε φ

ρ

4 +

⎡ ⎤

= ⎢ + ⎥

∫ ∫

( )

0 0

2 z r z

= ε ⎢ + − ⎥

⎣ ⎦

'

( )

2 2 2

1

2 2 2

2 2

cf) ', ' substitute '

' ' 0

r dr z r k

z r

k k

+ =

+

⎫

∫

( )

( )

( ^{2 2 2})¹

2 2

2 2 2 2 2 2

1 2 2

' 0,

' ,

2 ' ' 2

z b

z

r k z k z

r b k z b k z b k dk dk k

d k dk k

1 +

= = → = ⎫

⎪⎪

= = + → = + ⎬⇒ = =

⎪⎪

∴ ∫ ∫

( )

2 'r dr'= 2k dk ⎪ k

⎪⎭

Ex 3 9 A uniformly charged disk Ex 3-9 A uniformly charged disk

⎧ ⎡

(

^{2 2}

)

1¹² ⎤

0

1

ˆ 1 , 0

ˆ 2

z s z z b z

E V z V

z

ρ ε

⎧ ⎡ − ⎤

− + >

⎪ ⎢ ⎥

∂ ⎪ ⎣ ⎦

= −∇ = − ∂ = ⎨⎪ ⎡

(

^{2 2}

)

¹² ⎤

0

ˆ 1 , 0

2 z s

z ρ z z b z

ε

∂ ⎪−⎪⎩ ⎡⎢⎣ + + − ⎤⎥⎦ <

2 2

0

ˆ , 0

ˆ 4

For very large ^s

Note

z Q z

b z

z E z π ρ πε

⎧ >

⎪⎪⎨

2 0

2 0

For very large ,

4 ˆ , 0

4 z E z s

z Q

z z

z πε

πε

= = ⎨

⎪ − <

⎪⎩

3 6 Conductors in Static Electric Field 3-6 Conductors in Static Electric Field

Conductor : Free Electrons

The electrons in the outermost shells of the atoms of conductors are free to move

Insulator or dielectrics

The electrons in the atoms of insulators are confined to their orbits: no free electrons

electrons

Semiconductors

A relatively small number of freely movable charge carriers A relatively small number of freely movable charge carriers

→ Allowed energy bands for electrons.

→ Forbidden band → Band gap

3 6 Conductors in Static Electric Field

Charge introduced → Field will be set up → Charges will move away

3-6 Conductors in Static Electric Field

Charge introduced Field will be set up Charges will move away

from each other to reach conductor’s surface until the charge and the field inside vanish

: Inside a conductor ρ =0, E =0

Under static conditions, the E field on a conductor surface is everywhere normal to the surface → Equipotential surface (No tangential component)

0 ( E inside conductor is zero)

abcdaE dl= Δ =Et w

∫

^{i ∵}

0

abcda

t s n

E E ds E s ρ s

∴ =

= Δ = Δ

∫

∫

ⁱ

0

0 s n

s

En

ε ρ

= ε

∴

∫

3 6 Conductors in Static Electric Field

Boundary conditions at a conductor / Free space interface

3-6 Conductors in Static Electric Field

Boundary conditions at a conductor / Free space interface

t 0 E

ρ

⎧ =

⎪⎨ (∵E field = 0 inside conductor)

0 s

En ρ

⎨ = ε

⎪⎩

(∵E field = 0 inside conductor)

Uncharged conductor in a static field

Electrons move in a direction opposite to that of the field

- Electrons move in a direction opposite to that of the field - Positive charges move in the direction of the field

- So that induced free charge will distribute on the conductor surface and create an induced field in such a way to cancel the external field inside the conductor and induced field in such a way to cancel the external field inside the conductor and tangent to its surface

Ex 3 11 A point charge +Q at the center of a conducting shell Ex 3-11 A point charge +Q at the center of a conducting shell

Determine E and V as functions of the radial distance R

0

Determine and as functions of the radial distance "Ungrounded"

(a) Spherical Symmetry _R ˆ

E V R

R > R ∵ E = E R

( )

2 1

0

1 2 1 1

4

,

S R

R

R R

E ds E R Q

Q Q

E V E dR

π ε

⋅ = =

= = −

∴ =

∫

∫ ^{( )}

( )

1 2 1 1

0 0

0

4 , 4

b , Ins

R R

i

R R

R R R

πε ^∞ πε

<

<

∫

2

ide a conductor E =0 ∴E_R =0

0

2

2 1

0 0

, whole conducting shell is an equipotential body 4

R

R R

V V Q

πε R

= = =

An amount of negative charge equal to must be induced on the inner shell surface at in order to cancel out the field induced by positive charge _i at center.

Note

Q

R R Q

−

= +

This implies again that +Q must be induced on the outer shell surface at R = R₀.

Ex 3 11 A point charge +Q at the center of a conducting shell Ex 3-11 A point charge +Q at the center of a conducting shell

( )R < R

3 2 3 3

0 0

(c)

,

4 4

i

R R

R R

Q Q

E V E dR C C

R R

πε πε

<

= = −∫ + = +

3 2 0

3

0 0 0 0

where, at is equal to at

1 1 1 1 1

4 4

i

i i

V R R V R R

Q Q

C V

R R R R R

πε πε

= =

⎛ ⎞ ⎛ ⎞

= ⎜ − ⎟ ∴ = ⎜ + − ⎟

⎝ ⎠ ⎝ ⎠

0 ⎝ 0 i ⎠ 0 ⎝ 0 i ⎠

Electric field intensity and potential variations of A point charge +Q at the center of a conducting shell