Chapter 5 : Steady Electric Currents Static Electromagnetics, 2008 SpringProf. C.W. Baek & H. Kim...

Chapter 5 : Steady Electric Currents

Static Electromagnetics, 2008 Spring Prof. C.W. Baek & H. Kim

•Types of electric currents– Convection currents result from motion of electrons and/or holes in a vacuum or rarefied gas. (electron beams in a CRT,

violent motion of charged particles in a thunder storm). Convection current , the result of hydrodynamic motion involving a

mass transport, are not governed by Ohm’s law.

– Electrolytic current are the result of migration of positive and negative ions.

Atoms of the conducting medium occupy regular positions in a crystalline structure and do not move. Electrons in the inner shells are tightly bound to the nuclei are not free to move away. Electrons in the outermost shells do not completely fill the shells; they are valence or conduction electrons and are

only very loosely bound to the nuclei. When an external E field is applied, an organized motion of the conduction electrons will result (e.g. electron

current in a metal wire). The average drift velocity of the electrons is very low (10 -4~10-3 m/s) because of collision with the atoms,

dissipating part of their kinetic energy as a heat.

Electrolysis chemical decomposition

+ion -ion

ElectrolyteUsually a diluted

salt solution

current

- Conduction currents result from drift motion of electrons and/or holes in conductors and semiconductors.

Chapter 5. Steady Electric Currents



• Steady current density

– Electric currents : motion of free charges

– Current density : current per unit area

Consider a tube of charge with volume charge density v

moving with a mean velocity uu along the axis of the tube. Over a period t, the charges move a distance l = u t. The amount of charge that crosses the tube's cross-sectional

surface s' in time t is therefore

If the charges are flowing through a surface s whose surface

normal is not necessarily parallel to uu,

' ' 'v v vq v l s u s t

n̂

vq u s t

v

qI u s J s

t

�� 3 2 [C/m m/s=A/m ]J u

�� : (volume) current density

[A]S

I J d s ��

The total current flowing through an arbitrary surface S is

/ [A]=[C/s]I dQ dt2/ [A/m ]J dI ds



For metal,

For semiconductors : cf) resistivity :

Ohmic media : material following Ohm's law

e eJ E E ��

: point form of Ohm's law σ: conductivity (S/m)

( )e e h h e h 1 ( m)

12 12

1212

/

/

S

V E E V

I J d s JS J I S

V IJ E V I RI

S S

��

( )RS

The resistance of a material having a straight length , uniform

cross section area S, and conductivity :

Example 5-1.

In vacuum,

eF qE m a

u adt

– In conductors and semiconductors, electrons and/or holes can not be accelerated due to the collision.

– The drift velocity is proportional to the applied E field.

( : electron mobility)e eu E ��

3 2 [C/m m/s=A/m ]J u��



•Electromotive force (emf)Static (conservative) electric field :

This equation tells us that a steady current cannot be maintained in the same direction in a closed

circuit by an electrostatic field (Charge carriers collide with atoms and therefore dissipate energy

in the circuit).

This energy must come from a nonconservative field source for continuous current flow (e.g.

battery, generator, thermocouples, photovoltaic cells, fuel cells, etc.).

These energy sources, when connected in an electric circuit, provide a driving force to push a

current in a circuit : impressed electric field intensity EEi i .

– EMF of a battery : the line integral of the impressed field intensity EEi i from the negative to the

positive electrode inside the battery.

0C

E d ��

1

0C

J d

��

For an ohmic material :

1 1

2 2iE d E d= × =- ×ò ò

ur r ur rl lV

Insidethe source

2 1

1 20

CE d E d E d× = × + × =ò ò òur r ur r ur r

l l lÑOutside

the sourceInside

the source

1 1

12 1 22 2iE d E d V V V

�� V

E��

2 1

1 20iE d E d× - × =ò ò

ur r ur rl l

1

2

1

2 (inside source, for )i

V E d

V E d emf

��

��

Current flows from (-) to (+) inside source!



• Kirchhoff's voltage law

When a resistor is connected between terminal 1 and 2 of the battery to complete

the circuit : the total electric field intensity (EE + EEii) must be used in the point form

of Ohm's law.

If the resistor has a conductivity , length , and uniform cross section S, J = I / S.

– Kirchhoff's voltage law : Around a closed path in an electric circuit, the algebraic

sum of the emf’s (voltage rises) is equal to the algebraic sum of the voltage drops

across the resistances.

i iJ

J E E E E

��

�� R

1 1C

IJ d RI

S

�� V

1

2

1

i iC C C C

i

J d E E d E d E d

E d

��

��

V

(V)j k kj k

R I V



• Equation of continuity and Kirchhoff's current law

– Principle of conservation of charge : If a net current I flows across the surface out of (into) the

region, the charge in the volume must decrease (increase) at a rate that equals the current.

– Kirchhoff's current law : Algebraic sum of all the dc currents flowing out of (into) a junction

in an electric circuit is zero.

VVVS

dvt

dvJdvdt

d

dt

dQsdJI

)(A/m 3

tJ

: Equation of continuity

For steady currents, and therefore =0t

0J

��0 0 (A)jS

j

J d s I ��

( )

/ 0

J E Et

Et

��

��( / ) 3

0 (C/m )te

: relaxation time

decay to 1/e (36.8% value)

For a good conductor(e.g. copper), = 1.5210-19 [s]

Charge relaxation

For ac currents, and therefore 0t

0J

��0 0 (A)jS

j

J d s I ��

Really?

Quasi-static case (at low frequency =0)



• Power dissipation and Joule's law

– Power dissipated in a conducting medium in the presence of an electrostatic field EE Microscopically, electrons in the conducting medium moving under the influence of an

electric field collide with atoms or lattice sites Energy is thus transmitted from the

electric field to the atoms in thermal vibration.

The work W done by an electric field EE in moving a charge q a distance is

For a given volume V, the total electric power converted into heat is

In a conductor of a constant cross section, , with measured in the direction JJ.

Since V = RI,

0

limt

ww qE p qE u

t

��

: )(W/m 3JEdv

dP Power density under steady-current conditions

Joule’s law: (W) dvJEPV

dvJEdvuqNEpdPi

iiii

i

dsddv d

L SP Ed Jds VI

2 (W)P I R



• Boundary conditions for current density

– For steady current density J J in the absence of nonconservative energy sources

(1) Normal component : the normal component of a divergenceless vector field is

continuous.

(2) Tangential component : the tangential component of a curl-free vector field is

continuous across an interface.

01

0

0 0

dJJ

sdJJ

C

S

Differential form Integral form

)(A/m 221 nn JJ

2

1

2

1

t

t

J

J The ratio of the tangential components of J J at two sides of an interface is equal to the ratio of the conductivities.

[HW] Solve Example 5-3.



• Resistance calculations

We have calculated the resistance of a conducting medium having a straight length

, uniform cross section area S, and conductivity .

This equation can not be used if the S of the conductor is not uniform How can

we calculate the resistance?

– Procedures for resistance calculation

(1) Choose an appropriate coordinate system for the given geometry.

(2) Assume a potential difference V0 between the conductor material.

(3) Find EE within the conductor (by solving Laplace's equation and taking ).

(5) Find resistance R by taking the ratio V0 / I.

( )RS

S

L

S

L

sdE

dE

sdJ

dE

I

VR

SS

sdEsdJI (4) Find the total current I from

E V ��

cf) S

L

E d sQC

V E d

��

��

RC



• Example 5-6 : Resistance of a conducting flat circular washer

Sol.

(1) Choose a coordinate system : CCS

(2) Assume a potential difference V0.

(3) Find E E .

Boundary conditions are :

(4) Find the total current I.

(5) Find R.

V0+

-

2

1 220

d VV c c

d

0 02 2ˆ ˆ, V VV

V E Vr r

�� 0

0 at 0

at / 2

V

V V

0 0 02 2 2ˆ lnb

S a

V V hVdr bJ E I J d s h

r r a

��

(at = /2 surface)

ab

hI

VR

ln2

0

Chapter 5 : Steady Electric Currents Static Electromagnetics, 2008 SpringProf. C.W. Baek & H. Kim...

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