Chapter 21. Current and Direct Current Circuits 21.1 Electric Current

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Ambrose Mitchell
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1 Chapter 21 Current and Direct Current Circuits 21.1 Electric Current Electric Current 1 is defmed as the rate ofcharge flowing through a cross-section. =dq dt The "81" unit ofelectric current is the Ampere or the Amp. la = 1 C/sec. The convention is that the direction of the electric current is opposite to the direction of flow of the "free electrons". A. Microscopic Model ofcurrent Let n = number of free electrons per unit volume q = charge of each electron (also called "e") A = cross-sectional area through which the free electrons flow. Vd = drift velocity of the free electrons (-- 1 mm/sec) lave = average current 1

2 1Q lave = 1t but lax 1t V d ave = {n q Vd } A Define the electric current density J as E Resistance and Ohm's Law n electrostatics, we learned that the electric field E inside a conductor in static equilibrium is zero. Connecting a battery (or potential difference) across the ends of a conductor sets up an electric field E inside the conductor. An electric current density J and an electric field E are established in a conductor whenever a potential difference is maintained across the conductor..,r ~e -~')oj'" (Ohm's Law) 2

3 J=oE -=-- where 0= electrical conductivity p= electrical resistivity ( in Ohm meters) a Consider the following conductor connected to a battery: --e >i,\ ~~~~ Applying Ohm's law yields: A 1 AV P One then defines the electrical resistance R as so that Ohm's law becomes, when applied to a resistor, AV=R ~ ~ ~~-- The "S" unit of electrical resistance is the Ohm (Q). 3

4 B. Change in Electrical Resistance with Temperature The electrical resistivity pof a metal varies linearly with temperature (for high temperatures) according to where Po == resistivity at temperature To P = resistivity at temperature T a == temperature coefficient of resistivity p OC T (i) high temperature resistivity (linear regime) due to collisions between the free electrons and metal atoms. (ii) low temperature resistivity (non-linear regime) due to collisions between free electrons with impurities and imperfections. Since p is proportional to R, then 4

5 21.3 fuwerconductors R(Q) 0.15 For superconductors the electrical resistivity drops to zero at a temperature known as the critical temperature T c.,/ i.,/ Te , Critical Temperatures for Various Superconductors Material T e (K) HgBa2Ca2CuSOS 134 1GURE 21.9 ~ Resistance versus l1-ba-ca-cu-o 125 temperature for a sample of mercury. Bi-Sr-Ca-Cu-O 105 The graph follows that of a normal YBa2Cus07 92 metal above the Clitical temperature NbsGe 23.2 T e The resistance drops to zero at T", NbsSn which is 4.2 K for mercury. Nb 9.46 Ph 718 Hg 4.15 Sn 3.72 Al 1.19 Zn 0.88 T(K) Some ceramics have a high T c (like Yba2Cu307) Superconducting metals have low values of T c. Copper, Silver, and Gold (which are excellent conductors) do not exhibit superconductivity. Superconducting magnets produce magnetic fields lox greater than those produced by the best electromagnets. Superconducting magnets are used in medical magnetic resonance imaging (MR) units, which produce high quality images of internal organs without excessive exposure of patients to x-rays or other harmful radiation Microscopic Model of Electrical Conduction 5

6 The electrical conductivity a depends on microscopic parameters such as (Drude model 1900). 2 ne L 0=- n = number offree electrons per unit volume ( ) e = charge of electron L = average time between collisions of free electrons with the metal atoms (also called lattice atoms). me = mass of electron == mean-free path == average distance traveled by the free electrons between collisions. V th == V == average speed of free electrons between collisions. (~ 10 6 m/s) _ v =V = th L 21.5 Electrical Power and Enera Electrical power is defmed as the rate of electrical energy supplied by a voltage source (battery), or the rate of electrical energy dissipated (converted from electrical 6

7 energy to other forms of energy such as thermal energy) by a resistor) is given by lp = (AV] For a resistor, AV = R, so that P'esistor = (R ) ~esistor 2 (AVY = R = ---l R 21.7 Resistors Connected in Series and in Parallel A. Resistors connected in Series 1 1 =1 2 =1 R 1 R 2 a b c t ~1, t. V J + - R<:q=R l +R 2 a ~~ vy L',V + 1' ~1 Where more generally, 7

8 The electric current is the same through all the resistors connected in series. That is and B. Resistors connected in parallel More generally, 6 V ~ Llv?: C, V R\ y y y T\ t R A-A a y y -T 2 rt + li~v = Req R 1 ~ ~ ".f T 8

9 The voltage is the same across all the resistors connected in parallel. and 21.8 Kirchhoff's Rules and Simple DC Circuits Consider the voltages across different circuit elements such as batteries and resistors. A. Batteries + [. (i) f YOU move across the battery from the positive terminal to the negative terminal, then the voltage across the battery is written as then ~ V = - E (ii) f YOU move across the battery from the negative terminal to the positive terminal, then the voltage across the battery is written as then ~ V = + E 9

10 B. Resistors ----~---- R.. (i) fyou move across the resistor in the same direction as the current through the resistor, then the voltage across the resistor is written as 11V = - R (ii) fyou move across the resistor in the opposite direction as the current through the resistor, then the voltage across the resistor is written as 11V = + R c. Kirchhoffs Junction Rule A statement of conservation of charge. A junction is where 3 or more wires connect. 2 ~nteri~g 2 ~ leaving JunctOn JunctOn currents 1 [currents \ 10

11 D. KirchhofCs Loop Rule A statement of conservation of energy. For any loop containing circuit elements, Go over problem 29 on page 797 ofthe textbook. Electromotive Force E Consider the circuit shown below which consists of a Battery The resistance r is called the internal resistance ofthe battery. Terminal b is maintained by the source at a higher potential than terminal a. The electromotiveforce E ofthe bqttery is defined as the work per unit charge performed by the source when a charge moves from the negative terminal a to the positive terminal b. prefer to think of the E as the voltage across the terminals ofthe battery 11

12 when the circuit is open, that is, when there is no current through the battery. Let's calculate the open-circuit terminal voltage AVba = Vb - Va, ofthe battery by starting at point a and moving to point b across the battery: AVba,= E- r Notice that ifthe current in the battery is zero, then the terminal voltage across the battery equals the emf E ofthe battery. fthe current through the battery is not zero, then the terminal voltage across the battery will be smaller than the emjeofthe battery. By the way, what ifthe above circuit a stronger source is connected in series with the load resistor R such that the current in the circuit is in the opposite direction shown? What is then the terminal voltage AVba = Vb - Va across the battery? n this case AVba = E+ r and notice that the terminal voltage across the battery will be higher than the emfeofthe battery. More generally, 12

13 Operation of a Three- Way Lightbulb: Wiring Diagram for a Household Circuit: ---, V Live Neurral \ S \~G+nc4 A-ff ~ i ayl[ e s TQO\~-\-er Ro.*J cjc qs-o w4 - b,..\j \'2.D YD Hs ---- D - C\ ~ Wi.\m.L---L-~. = l\~l A~1~ 5 ~Rl--'-----r--'~,---R2---'--~R_ G) me 0 V (0 t~ ONlu. r ~ ~ Bc\~~. EexGh Figure:aellM Wirin-g,tliagraro for a household circuit. The resistances Gd'c b D 0 wc01j represent appliances or other electrical devices that operate with ) 'ni\~ -:::.. w~tts - ~" 12-0 vo\1s an applied voltage of 120 V. _ P "f) D ::. 'S A","",\,5 ) Hov\r d.rj-t:c (0 bvio\a.sl..,~ \ ~ ~ W~S J r ==-L =-llooo ~~ ~'- La y~ "'L. ::: \2'. ~ Amys ' 13

PHY102 Electricity Course Summary

PHY102 Electricity Course Summary TOPIC 1 ELECTOSTTICS PHY1 Electricity Course Summary Coulomb s Law The magnitude of the force between two point charges is directly proportional to the product of the charges and inversely proportional