Steady Electric Currents

(1)

Field and Wave Electromagnetic Chapter.5

(2)

Introduction Introduction

9 Stead c ents

9 Steady currents

¾ Conduction current

¾ Electrolytic current

¾ Convection current

(3)

Current Density and Ohm’s Law Current Density and Ohm s Law

N : number of charge carrier / unit volume N

u

t u t

Δ Δ

c

: number of charge carrier / unit volume : velocity

During time interval , distance of charge carrier movement : Δ s

d

g , g

The amount of charge passing through the surface area Δ = Q Nqu n s t i Δ Δ

I Q Nqu n s J s t

J N N

Δ = Δ = Δ = ⋅ Δ

∴ Δ i

and , where : volume charge density

s

J Nqu Nq

I J ds N

ρ

= =

= ∫ ⋅

and where : volume charge density

f) Nq

J u

ρ

=

= cf)

then, : Convection Current density

(4)

Ex 5 1) Space charge limited vacuum diode Ex 5-1) Space-charge-limited vacuum diode

El t itt d f th d

0 V =

- Electrons are emitted from a cathode.

- at cathode.

- Electrons are collected by an anode V = V

0

Electrons are collected by an anode.

- at anode.

- Electrons at cathode has a zero initial velocity.

0

.

J V

y

Find the relation between and

(5)

sol)

0

ˆ ˆ ( )

(0)

_y

(0) 0

y

E yE y dV y

dy

₌

= = − =

sol) c

0 y

In the steady state, the current density is constant

d i d d t f

d

ˆ ˆ ( ) ( ) ( ) 0

y J yJ y y u y

y ρ ρ

= − =

< ∵ and independent of

where ( ) y 0 ( electrons)

u

ρ < ∵

where, ( electrons)

e ( ) y and E y ( ) is governed by Newton's law.

( ) ( )

( ) dV y

m du y eE y e

dt dy

⋅ = − =

(6)

2

1 2 1

du du dy du d

m m mu mu

dt dy dt dy dy

d dV

⎛ ⎞

= = = ⎜⎝ ⎟⎠

⎛ ⎞

⎜ ⎟

2

(0) 0 (0) 0 0

mu e

dy dy

u V y

∴ ⎛⎜⎝ ⎞ =⎟⎠

= = =

Integrating both side with a given boundary condition at

2

1

(0) 0 (0) 0 0

u V y

mu eV

= = =

=

Integrating both side with a given boundary condition , , at 1

2

1

2 2

e :

u V

m

⎛ ⎞

= ⎜⎝ ⎟⎠

analogy to free fall motion in the gravitational field

(7)

Ex 5 1) Space charge limited vacuum diode Ex 5-1) Space-charge-limited vacuum diode

( )

( ) V y ρ V y

To find , we have to solve Poisson's equation with expressed in terms of

1 2

, ( 0)

J u

J m

J V

ρ ρ

ρ

⁻

= − <

= − = −

2

u e

ρ

Poisson's

9 Poisson s equation

0

V ρ

∇

²

= − ε equation

2

1/ 2 2

0 0

2

d V J m

dy e V

ρ ε ε

= − =

−

∴

(8)

2 1

2

2 2

d V dV dV

dy dy K V dy

⎛ ⎛ ⎞

−

⎜ ⋅ ⎜ ⎟ = ⋅ ⋅

⎜ ⎝ ⎠

⎜

₂

1

4

2

d dV d

K V

dy dy dy

⎜ ⎝ ⎠

⎜ ⎡ ⎛ ⎞ ⎤ ⎡ ⎤

⎜ ⎢ ⎜ ⎟ ⎥ = ⎢ ⋅ ⎥

⎜ ⎢ ⎣ ⎝ ⎠ ⎥ ⎦ ⎣ ⎦

⎜ cf)

2 1

4

2

dV K V c

dy

⎣ ⎦

⎜ ⎜

⎛ ⎞

⎜ ∴ ⎜ ⎟ = ⋅ +

⎜ ⎝ ⎠

⎝

2 1

4

2

dV J m

V c

⎝

⎛ ⎞

= +

∴ ⎜ ⎟

0

2 V c

dy = ε e +

⎜ ⎝

∴ ⎟ ⎠

(9)

0 0 dV 0

B C t V ( i iti l l it )

1

0, 0, 0

0

y V

dy c

= = =

∴ =

B.C. at (zero initial velocity)

1 1

4 4

0

1 1

2 2

dV J m

dy

ε

e V

⎛ ⎞

∴ = ⋅⎜ ⎟

⎝ ⎠

⎛ ⎞

1 4

4

0

3 1

4

2 2

0 0

4 J m

V dV dy

e

y V

J

ε

− ⎛ ⎞

∴ = ⎜⎝ ⎟⎠

⎛ ⎞ ⎧

at

3 4

4 0

0 0

0, 0

4 2 . .

3 2 ,

y V

J m

V d

y d V V

ε

e

= =

⎛ ⎞ ⎧

= ⎜⎝ ⎟⎠ ⎨⎩ = =

∴ at

B.C

at

4 ⁰₂ 2 ₀³² 9

J eV

d m

∴ =

ε

⇒

Child Langmuir law

(10)

9 In case of conduction currents

J = ∑ N q u

9 In case of conduction currents.

- electrons, holes

i i i

i

J = ∑ N q u

- Atoms remain neutral

A d ift l it i di tl ti l t th l t i fi ld

- Average drift velocity is directly proportional to the electric field intensity

- metallic conductors

, :

e e

u = − μ E μ

²

⋅

- metallic conductors

: velocity mobility (m /V S)

(11)

3 4 3

: 3.2 10 : 1.4 10 : 5 2 10

μ

⁻

−

×

cf) e for Copper for Al

for Silver : 5.2 10

e e , e

J

ρ μ

E

ρ

Ne

×

= − = −

for Silver

where : charge density of drifting electrons.

_e _e

h h

J

σ

E

σ ρ μ σ ρ μ ρ μ

= = −

= − + ⇒

,

cf) for semiconductor

Point function conductivity for metallic conductor

0.38, 0.18

e e h h

e h e h

σ ρ μ ρ μ

μ μ μ μ

+ ⇒

≠ = =

cf) for semiconductor

cf) In general , for germanium,

for silicon,

μ

_e =0.12,

μ

_h = 0.03

electrons holes electrons holes

(12)

Current Density and Ohm’s Law Current Density and Ohm s Law

9 Conductivity

J = σ E 9

¾

Conductivity

Isotropic materials obeying, : ohmic media Conductivity

( )

( ) ( )

( )

σ ⋅

×

⁷

Co duct ty

: A/ V m or Simens per meter S/m for copper : 5.80 10 S/m

⎧ ⎪ ( )

σ

for copper : 5.80 10 S/m for germanium :

cf) is ( )

( )

10

⁻3

⎧ ⎪

⎪ ⎨ ×

⎪ ⎪

2.2 S/m for silicon : 16 S/m

( )

⎪⎩ for Hard rubber : 10

^-15

S/m

(13)

9Ohm's law

12

J E

V RI

σ

= ⇒

9Ohm s law

point form of Ohm's law cf) Not a point relation

σ

,

- Assume a piece of homogeneous conducting material with

conductivity length l, S

V

uniform crosssection .

12 12

s ,

V El E V

l I J ds JS J I

S

= ⇒ =

=

∫

⋅ = =

12

s S

V I

S l

l

σ

= ⋅

⎛ ⎞

∫

12

V l I R I

σ

S

⎛ ⎞

=⎜⎝ ⎟⎠ =

∴

(14)

Electromotive Force and Kirchhoff’s Voltage Law Electromotive Force and Kirchhoff s Voltage Law

C E dl⋅ = ⇒0

∫

9 Static E-field is irrotational (i.e. Conservative)

1 0

C

J E

J dl

σ

=

⋅ =

∴

∫

For an ohmic material,

Steady current cannot be maintained in the same

:

di ection in a closed ci

∫

C^Cσ direction in a closed ci c it b an elect ostatic field

∫

rcuit by an electrostatic field.

- To have steady current, nonconservative field should supply the energy which will be dissipated by collision

- The source of nonconservative field : chemical energy, generators.

- Impressed electric field intensity ⇓

⇓

driving forces for charge carrier

Ei

Equivalent impressed electric field intensity

(15)

Electromotive Force and Kirchhoff’s Voltage Law Electromotive Force and Kirchhoff s Voltage Law

E is produced by chemical action

(

ⁱi

)

E E

is produced by chemical action : impressed electric field intensity

Ei = −E

∵ ⇒ No current flows in the open-circuited battery

the net force acting on

⎛⎜

⎝ the charge carriers must vanish

1 1

2 E dli ⋅ = − 2 E dl⋅ =

∫ ∫

Inside the source

: electromotive force

d i i f f h i

V

⇒

driving forces for charge carrier

Ei

9 Electro motive force : The line integral of the impressed field intensity g p y _i from the negative to positive electrode inside the battery

(16)

Electromotive Force and Kirchhoff’s Voltage Law Electromotive Force and Kirchhoff s Voltage Law

9

Conservative electrostatic field

2 1

1 2 0

C E dl⋅ = E dl⋅ + E dl⋅ =

∫ ∫ ∫

outside the inside the source source

Conservative electrostatic field

∫

2 E dl⋅

outsi

∫

2

V = 1 = V₁₂ = V₁ −V₂

de the source

: voltage rise

With a resistor connecting two terminals the total electric field intensity

(E +Ei)

With a resistor connecting two terminals, the total electric field intensity must be used in the point form of Ohm's law

J =

σ

((E + Ei),),

Ei E

exist inside the battery only, exist both inside and outside the battery

(17)

Electromotive Force and Kirchhoff’s Voltage Law Electromotive Force and Kirchhoff s Voltage Law

1

J ^,

( )

¹

1 0

i i

C

E E J E E dl J dl RI

J dl

σ σ

+ = + ⋅ = ⋅ =

⋅ =

∫ ∫

∫

cf) no source of nonconservative field

V = C

C 0

k k

J dl

R I

σ

⁼

=

∫

∑ ∑

cf) no source of nonconservative field

: Kirchhoff's voltage law

Vj _{k k}

j k

∑ ∑

R I

: Kirchhoff s voltage law

Vj

→

Around a closed path in an electric circuit, the algebraic sum of the emf's (voltage rises) is equal to the algebric sum of the voltage the emf s (voltage rises) is equal to the algebric sum of the voltage drops across the resistances.

(18)

Equation of Continuity and Kirchhoff’s current law Equation of Continuity and Kirchhoff s current law

S V

dQ d

I J ds dv

dt dt ρ

=

∫

⋅ = − = −

∫

9 The principle of conservation of charge

S

total charge dt dt

total outward flux ⇒Decrease in total charge inside V

total charge Q

J dv V dv t

ρ

⎛

ρ

∇ ⋅ = − ∂∂ ⎜⎝

∫ ∫

¾

V

Applying divergence theorem, and assuming the stationary volume taking partial derivative in the integral since

,

may be a function of time as well as space

V ∂t ⎜⎝

∫

V

∫

may be a function of time as well as space

J

t

ρ

∇ ⋅ = −∂ ⇒

∴ ∂

: continuity equation point relationship

ρ

0

⇒ ∂ =

Steady current Charge density does not vary with time : i.e.

(19)

Equation of Continuity and Kirchhoff’s current law Equation of Continuity and Kirchhoff s current law

In other words,

0 0

J d I

⇒

⇒ ⇒

∫

,

Steady current = Divergenceless = Solenoidal

The field lines ( or stream lines) of steady current close upon themselves.

i e

∫

_S ^{J ds}^⋅ ^{= ⇒}⁰

∑

^I^j ^{= ⇒}⁰ Kirchoff's curren^{t law}

i.e. Kirchoff's curren

j

ρ

∂

∑

^{t law}

cf) Relaxation time

,

J J E

t

E E

t

ρ σ

ρ ρ

σ ε

∇ ⋅ = −∂ =

∂

∇ ⋅ = −∂ ∇ ⋅ =

∂

, and

and in a simple medium 0

t t

ρ σ ρ ε

∴ ε

∂

∂ + = ∂

σ t

⎛ ⎞⎜ ⎟

₀

0

t

e

t ρ ρ ε

ρ τ ε

−⎜ ⎟⎝ ⎠

=

= =

∴

,

where initial charge density at , relaxation time =

(20)

Power Dissipation and Joule’s law Power Dissipation and Joule s law

Macroscopic observation 9

E ⇒ ⇒

⇒

⇒ p

drift motion of conduction electrons electrons collide with atoms vibration of lattice

i e Electric field energy Thermal vibration

w E

⇒ Δ

i.e. Electric field energy Thermal vibration

The work by o q

w qE l Δ = ⋅ Δ n a charge

lim0 :

t

p w qE u u

t dv

Δ →

Δ = ⋅

∴ Δ

Power = where, drift velocity Total power in a volume

( )

i i i i

i i

dP p E N q u dv E J dv dP

⎛ ⎞

= = ⋅⎜ ⎟ ⋅

⎝ ⎠

∑ ∑

3

=

Power density under steady

( )

dP E J dv

v

= ⋅ ³ Power density under steady

or W/m :

current conditions For a given volume

(21)

Power Dissipation and Joule’s law Power Dissipation and Joule s law

f) S i l

dv = ds dl dl J

cf) Special case

Conductor : Constant cross section

measured in the direction

2

,

L s

dv ds dl dl J

P E dl J ds VI I R

=

= ∫ ∫ = =

measured in the direction

(W)

∫

f)

:

V E dl

E

= − ∫

cf)

note Justification of in a conductor c

d

Voltage rise : Source of nonconservative field Not P.E.C : relaxation time

ρ e ∂

Steady current driven by non 0

t ρ σ

⇒ − ∂ =

∂

⇒ ⇒ = ∞

f

conservation source

resistance finite , P.E.C

(22)

Boundary Conditions for Current Density Boundary Conditions for Current Density

For steady current,

0, 0

0 1 0

J S J ds

J J dl

∇ ⋅ = ⋅ =

∇×⎜ ⎟⎛ ⎞

∫

For steady current,

: continuity equation

: conservative electrostatic field

0, 0

C J dl

σ σ

∇×⎜ ⎟ = ⋅ =

⎝ ⎠

∫

: conservative electrostatic field The normac

J J

∴ =

l component of a divergenceless vector field is continuous

1n 2n

J J

∴ =

The tangential component of a curl-free vector field is continuous across an

d

interface

1 2 1 1

1 2 2 2

t t t

t

J J J

J

σ

σ ⁼ σ ⁼σ

or

Boundary conditions between two different lossy dielectrics e ε₁ ε₂,

permitivities and

(

finite conductivity and ^σ1 ^σ2

)

- The tangential component of the electric field

(23)

Boundary Conditions for Current Density Boundary Conditions for Current Density

1 2 1 2 2

1 2

n n n n

J J E E

D D

σ σ

ρ

= ⇒ =

− =

1

- Normal component of the electric field

(

^D¹ⁿ ^D²ⁿ

ρ

^s ⁿ²

)

with the reference with normal from medium 2 to medium 1

1 2 2

E E

ε ε ρ

⇒ ₁ − =

₁ ₂ ₂

2 1

n n s

E E

ε ε ρ

σ σ

ρ ε ε ε ε

⇒

⎛ ⎞ ⎛ ⎞

=⎜ − ⎟ =⎜ − ⎟

∴

1

1 2 2 1 2 1

1 2

s E n E n

ρ ε ε ε ε

σ σ

σ ε

=⎜ ⎟ =⎜ ⎟

⎝ ⎠ ⎝ ⎠

∴

2 2

1 1

, _S 0

σ ε ρ

σ

⁼

ε

⁼

if then

(24)

Resistance Calculation Resistance Calculation

D ds E ds

Q ∫

s

⋅ ∫ ε ⋅

L L

C Q

V E dl E dl

= = =

− ⋅ − ⋅

∫ ∫

9

When the dielectric medium is lossy, a current will flow from the positive to the negative conductor.

J σ E J E

⇒ =

and and are i

dl dl

∫ ∫

n the same direction

L

E dl

L

E dl

V C

R RC

I J ds E ds G

ε σ σ

− ⋅ − ⋅

= = = ⇒ = =

⋅ ⋅

∴ ∫ ∫

∫ ∫