Chapter 5 Direct Currents in Electric Circuits Conduction of Electricity Ohm’sLaW Non-linear resistors Resistors connected Kirchhoffs laws Substitution theorem Mesh current
Chapter 5 Direct Currents in Electric Circuits ◼ Conduction of Electricity ◼ Ohm’s Law ◼ Non-linear Resistors ◼ Resistors Connected ◼ Kirchhoff’s Lawss ◼ Substitution Theorem ◼ Mesh Current
Voltage and current Sources Thevenin theorem a Transient in rc circuits If the electric field e is maintained in conductor by an external source say, a battery, then charges drift in the field and there is an electric current
◼ Voltage and Current Sources ◼ Thevenin Theorem ◼ Transient in RC Circuits If the electric field E is maintained in conductor by an external source, say, a battery, then charges drift in the field and there is an electric current
5. 1 Conduction of Electricity The current density vector J Its magnitude is the amount of charge flowing thru a surface in unit time, and has unit ampere second. meter 2,O meter It points in the direction of flow of positive charge If negative charges flow, such as conduction electrons in a conductor, then j points in the opposite direction of flow of negative charges. (See fig 5-1 Seme-conductors contain one or two types of mobile charge, namely electrons and holes For ordinary conductors, it holds that J=gE
Here o is the conductivity of the material. (see Table 5-1 for various common materials Microscopically speaking, current in conductor forms due to the drift of conduction electrons = nev ere n-- the conduction electron density, the magnitude of the electron charge he drift velocity of the electron charge For example, in copper, n=8.5 102elctron/M3 if =100 A/M2, then the drift velocity 74×10-5M/s~026M/h 已 -the drift velocity is low different from thermal motion( 10-M/s
Conservation of Charge low consider a volume T bounded by a surface S inside conducting material. Let da be a small element of area on J. da is the charge flowing out thru da per unit time Then /J·da is the charge flowing out of s per unit time which is also the charge lost by the volume T per unit time dt L pO Thus, one has the conservation of charge d sJ·da= dt paT
Using the divergence theorem the l.h. s of the equation is written as a volume integration /sJda=V·Jdr, so the conservation equation is 1V·Jr=-mFpr. Since this is valid for an t is arbitrary This is the conservation of charge stated in differential orm
Homogeneous Conductor in Steady state If the system is in steady state, d=0, thus V·J=0. Then by j =oE, one has V·aE=0. If the conductor is also homogeneous o is independent ot position, one has V·E=0. Thus by v·E=p/∈0, i. e. under steady-state condition, the net charge den sity inside a homogeneous conductor carrying a current s zero
5.2 Ohm's law Given an element of conductor of length l and cross section a If a potential difference v is maintained btwn its two ends, there appears a current I From =oE with=I/A andE=v/l, one has a l so the current is given by the following where the resistance is A -Ohm's law
Remarks (1)The current thru a resistor is a the voltage across (2)A resistor satisfying this law is said to be linear 3 The current and voltage are measured as in fig. 5-4 The current thru a voltmeter is usually assumed to be negligible compares with that thru R R Figure 5.4 Measurement of the current I through a resistor R, and of the voltage across it. a circle marked I represents an ammeter, and a circle marked V,a voltmeter
The heat generated by the joule effect The work done on the carriers is Qv The work done on the carriers in unit time is QV t The work done on the carriers in unit time and unit volume is ViV 3-12l JE=O E2 This is converted into the heat generated in unit time and unit volume due to the joule effect or a resistor carrying a current I the voltage across it is IR and the heat generated in unit time is /V=/(IR)=R