Steady Electric Currents

8/8/2019 Steady Electric Currents

1/40

Steady Electric Currents


2/40

Current Density and Ohms Law

Consider the steady motion of one kind ofCharge carriers, each of charge q (which isvefor electrons), across an element surfaceswith

a velocity u, as shown in Fig. 51. If N is thenumber of charge carriers per unit volume, thenin time t, each charge moves a distance u t,and the amount of charge passing through the

surfaces is:

Since current is the time rate of change of

2Steady Electric Currents


3/40


charge, we have:

In Eq. (52) we have written s=an

s as avector quantity. It is convenient to define a vectorpoint function, volume current density, orsimply, current density, J, in amperes per

squaremeter:

so that Eq. (52) can be written as:



4/40


The total current I flowing through an arbitrarysurface S is then the flux of the J vector throughS:

Noting that the product Nq is in fact free percharge volume, we may rewrite Eq. (53) as:

Which is the relation between the convectioncurrent density and the velocity of the chargecarrier.



5/40


In case of conduction currents there may bemore than one kind of charge carriers (electrons,holes and ions) drifting with different velocities.

Eq. (53) should be generalized to read:

The conduction currents are the result of the drift

motion of charge carriers under the influence ofan applied electric field. The atoms remainneutral ( = 0). It can be justified analytically thatfor most conducting materials the average drift



6/40


velocity is directly proportional to the electricfield intensity. For metallic conductors we write:

where e is the electron mobility measured in(m2/Vs). It is 1.4 104 (m2/Vs) for aluminumand 5.2 103 (m2/Vs) for silver. From Eqs. (5

3) and (519) we have:

where e=Ne is the charge density of thedrifting electrons and is a ve quantity.



7/40


Eq. (520) can be rewritten as:

where the proportionality constant = e

e

, isa macroscopic constitutive parameter of themedium called conductivity.

For semiconductors, conductivity depends on

the concentration and mobility of both electronsand holes:



8/40


where subscript h denotes a hole. In general, e h. For germanium, typical values are e =0.38, h = 0.18; for silicon, e = 0.12, h = 0.03.

(m2/Vs). Eq. (521) is a constitutive relation of a

conducting medium. Isotropic materials for whichthe linear equation Eq. (521) holds are calledohmic media. The unit for is ampere per voltmeter (A/Vm) or siemens per meter (S/m).Copper, the most commonly used conductor,



9/40


has a conductivity 5.80 107 (S/m). On the otherhand, the conductivity of germanium is around2.2 (S/m), and that of silicon is 1.6 103 (S/m).

The conductivity of semiconductors is highlydependant of (increases with) temperature. Hardrubber, a good insulator, has a conductivity ofonly 1015 (S/m). However, note that, unlike the

dielectric constant, the conductivity of materialsvaries over a very wide range. The reciprocal ofconductivity is called resistivity, in ohm-meters



10/40


(m). We prefer to use conductivity; there isreally no compelling need to use bothconductivity and resistivity.

We recall Ohms Law from circuit theory that thevoltage V12 across a resistance R, in which acurrent I flows form point 1 to point 2 is equal to ,that is:

Here R is usually a conducting material of givenlength; V12 is the voltage between two terminals



11/40


1 and 2; and I is the total current flowing fromterminal 1 to terminal 2 through a finite cross-section.

Eq. (523) is nota point relation. Although thereis little resemblance between Eq. (521) andEq. (523), the former is generally referred toas the point form ofOhms law. It holds at allpoints in space, and can be a function ofspace coordinates.

Let us use the point form of Ohms law to derive



12/40


13/40


The total current is:

or:

Using Eqs. (524) and (525) in Eq. (521),we obtain:

or:



14/40


which is the same as Eq. (523) From Eq. (526) we have the formula for resistance of astraight piece of homogeneous material of a

uniform cross section dor steady current (d.c.):

We could have started with Eq. (523) as the

experimental Ohms law and applied it to ahomogeneous conductor of length land uniformcross-section S. Using the formula in Eq. (527), we could derive a point relationship in Eq.



15/40


(521).

The conductance, G, or the reciprocal ofresistance, is useful in combining resistances in

parallel. The unit for conductance is (1), orsiemens (S):

From circuit theory we know the following:a) When Resistances R1 and R2 are connected in

series (same current), the total resistance R is:



16/40


b) When resistances R1 and R2 are connected inparallel (same voltage), we have:

c) or:



17/40


18/40

Electromotive Force and Kirchhoffs

Voltage Law

direction in a closed circuit by an

electrostatic field. A steady current in a circuitis the result of motion in charge, which, in their

paths, collide with atoms and dissipate energy inthe circuit. This energy must come from anonconservative field, since a charge carriercompleting a closed circuit in a conservative field

neither gains nor loses energy. The source ofthe nonconservative field may be electricbatteries (conversion of chemical energy toelectric energy), electric generators (conversion



19/40


Voltage Law

of mechanical energy to electric energy),thermocouples (conversion of thermal energy toelectric energy), photovoltaic cells (conversion of

light energy to electric energy), or other devices.These electrical energy sources, whenconnected in an electric circuit, provide a drivingforce for charge carriers. This force manifests

itself as an equivalent impressed electric fieldintensityEi.

Consider an electric battery with electrodes 1



20/40


Voltage Law

and 2, shown schematically in Fig. 54.Chemical action creates a cumulation of +ve andve charges at electrodes 1 and 2, respectively.

These charges give rise to an electrostatic fieldintensity E both inside and outside the battery.Inside the battery, E must be equal in andopposite in direction to the nonconservative Ei

produced by chemical action, since no currentflows in the open-circuited battery and the netforce acting on the charge carriers must vanish.The line integral of the impressed field intensity



21/40


Voltage Law

Ei from the ve to the +ve electrode(fromelectrode 2 to electrode 1 Fig. 54) inside thebattery is customarily called the electromotive

force (emf) of the battery. The SI unit for emf isvolt, and an emf is not a force in newtons.Denoted by , the electromotive force is themeasure of the strength of the nonconservative

source. We have:

The conservative electrostatic field intensity E



22/40


Voltage Law

satisfies Eq.(5-31)

Combining Eqs.(5-33) and (5-34), we have:

Or

In Eqs.(5-35) and (5-36) we have expressed theemf of the source as a line integral of theconservative E and interpreted it as a voltagerise. In spite of the nonconservative nature of E

i

,22Steady Electric Currents


23/40


Voltage Law

the emf can be expressed as a potentialdifference between the +ve and ve terminals.This was what we did in arriving at Eq.(5-24).

When a resistor in the form of Fig. 5-3 isconnected between terminals 1 and 2 of thebattery, completing the circuit, the total electricfield intensity (electrostatic E caused by charge

cumulation, as well as impressed Ei caused bychemical action), must be used in the point formof Ohms law. We have, instead of Eq.(5-21),



24/40


Voltage Law

where Ei exists inside the battery only, while Ehas a nonzero value both inside and outside the

source. From Eq.(5-37) we obtain:

The scalar line integral of Eq.(5-38) around the

closed circuit yields, in view of Eqs.(5-31) and(5-33),



25/40


Voltage Law

Eq.(5-39) should be compared to Eq.(5-32),which holds when there is no source ofnonconservative field. If the resistor has a

conductively , length l, and uniform crosssection S, J=I/S and the right side of Eq.(5-39)becomes RI. We have:

If there are more than one source ofelectromotive force and more than one resistor(including the internal resistances of the



26/40


Voltage Law

sources) in the closed path, we generalizeEq.(5-40) to:

Eq.(5-41) is an expression of KirchhoffsVoltage Law. It states that, around a closed

path in an electric circuit, the algebraic sumof the emfs (voltage rises) is equal to the

algebraic sum of the voltage drops across

the resistances. It applies to any closed path in



27/40


Voltage Law

a network. The direction of tracing the path canbe arbitrarily assigned, and the currents in thedifferent resistances need not be the same.

Kirchhoffs voltage law is the basis for loopanalysis in circuit theory.



28/40


29/40

Equation of Continuity andKirchhoffs Current Law

region, the charge in the volume must increaseat a rate equal to the current. The currentleaving the region is the total outward flux of the

current density vector through the surface S. Wehave:

Divergence theorem, Eq. (2115), may beinvoked to convert the surface integral of J to thevolume integral of J. We obtain, for astationary volume:



30/40


In moving the derivative of inside the volumeintegral, it is necessary to use partial

differentiation because may be a function oftime as well as of space coordinates. Since Eq.(543) must hold regardless of the choice of V,the integrands must be equal. Thus we have:

This point relationship derived from the principleof conservation of charge is called the equation



31/40


of continuity.

For steady currents, charge density does notvary with time, p/t= 0. Eq. (544) becomes:

Thus steady electric currents are divergencelessor solenoidal. Equation (545) is a point

relationship and holds also at points where =0(no flow source). It means that the field lines orstreamlines of steady currents close uponthemselves, unlike those of electrostatic field



32/40


33/40


Kirchhoffs current law is the basis for nodeanalysis in circuit theory.

In Section 36, we stated that charges

introduced in the interior of a conductor willmove to the conductor surface and redistributethemselves in such a way as to make =0 andE=0 inside under equilibrium conditions. We are

now in a position to prove this statement and tocalculate the time it takes to reach anequilibrium. Combining Ohms law, Eq. (521),



34/40


with the equation of continuity and assuming aconstant , we have:

In a simple medium, E=/, and Eq. (548)becomes:

The solution of Eq. (549) is:

where 0 is the initial charge density at t=0. Both



35/40


and 0 can be functions of the spacecoordinates, and Eq. (550) says that thecharge density at a given location will increase

with time exponentially. An initial charge density0 will decay to 1/e or 36.8% of its value in atime equal to:

The time constant is called the relaxationtime. For a good conductor such as copper=5.80 107 (S/m), 0=8.85 10

12 (F/m)



36/40


equals 1.521019 (s), a very short time indeed.The transient time is so brief that, for all practicalpurposes, can be considered zero in the

interior of a conductorsee Eq. (369) inSection 36. The relaxation time for a goodinsulator is not infinite but can be hours or days.



37/40

Power Dissipation and Joules Law

Under the influence of an electric , conductionelectrons in a conductor undergo a drift motionmacroscopically. Microscopically, these

electrons collide with atoms on lattice sides.Energy is thus transmitted from the electric fieldto the atoms in thermal vibration. The work wdone by an electric field E in moving a charge q

a distance l is qE(l), which corresponds to apower:



38/40


where u is the drift velocity. The total powerdelivered to all the charge carriers in a volumedv is:

which, by virtue of Eq. (518) is:

or:

Thus the point function EJ is a power density



39/40


under steady-current conditions. For a givenvolume V the total electric power converted intoheat is:

This is known as Joules Law. (Note that the SIunit for P is watt, not joule, which is the unit forenergy or work.) Equation (553) is the

corresponding point relationship.

In a conductor of constant cross section, dv= dsdl, with dlmeasured in the direction J. Equation



40/40

Steady Electric Currents

Documents

Transcript of Steady Electric Currents