Preamble. Flow and Fluid Velocity. In this section of my lectures we will be. To do this we will use as an analogy
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- Clarissa Willis
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1 Preambe Resistance Physics, 8 th Edition Custom Edition Cutne & Johnson Chapter 20.3 Pages In this section of my ectures we wi be deveoping the concept of resistance. To do this we wi use as an anaogy water fowing in a pipe. (R. Boton - 202) Resistance (R. Boton - 202) Resistance 2 Physics 55.3: Introduction to Eectricity and Magnetism Physics 55.3: Introduction to Eectricity and Magnetism 2 Fow and Fuid Veocity Fuid in a pipe with cross sectiona area, A, and ength,, is pushed out by a punger that is moving at a constant veocity. It takes the punger t seconds to trave the ength of the pipe. voume A fow = = = A = Av time t t where: A is the cross sectiona area of the pipe., is the ength of the pipe. v is the veocity of the punger. (R. Boton - 202) Resistance 3 (R. Boton - 202) Resistance 4 Physics 55.3: Introduction to Eectricity and Magnetism 3 Physics 55.3: Introduction to Eectricity and Magnetism 4
2 Reating Current to Eectron Veocity Now imagine the punger is pushing ony the free eectrons in a copper wire (a conductor) and eaves behind the eectrons that are bound to the protons. Copper has 8.5x0 22 free eectrons per cubic cm, which in terms of charge is.36x0 4 C/cm 3. Reca that average current is Charge passing pane I = t So (assume an arbitrary pane ocation) I = Q t where 4.36x0 C/cm = t 3 A 4 =.36x0 C/cm 3 A v v is the veocity at which the eectrons move (drift) through the wire. (R. Boton - 202) Resistance 5 (R. Boton - 202) Resistance 6 Physics 55.3: Introduction to Eectricity and Magnetism 5 Physics 55.3: Introduction to Eectricity and Magnetism 6 Reating Eectron Veocity to Eectric Fied Appy votage V ab across the ends of the copper wire. Look at what happens to the free eectrons in the copper wire Then there wi be an eectric fied in the wire V E = ab (R. Boton - 202) Resistance 7 (R. Boton - 202) Resistance 8 Physics 55.3: Introduction to Eectricity and Magnetism 7 Physics 55.3: Introduction to Eectricity and Magnetism 8
3 The drift veocity is The average drift veocity of the eectron in the direction of the wire is proportiona to the eectric fied ee V vave τ c constant t E = cv E = cv 2 m where τ c is the time between coisions, is a constant with units of cm 2 c v V s (R. Boton - 202) Resistance 9 (R. Boton - 202) Resistance 0 Physics 55.3: Introduction to Eectricity and Magnetism 9 Physics 55.3: Introduction to Eectricity and Magnetism 0 Votage Current Reationship Substituting the expression for the average veocity into the previous equation for current yieds I = ( c.36x0 4 C/cm 3 ) v V A The constant term in brackets is σ (sigma). It has units of C. ab s V cm This constant is caed conductivity. Resistivity Rewrite the equation to isoate votage V ab = I = ρ I σ A A V cm A The constant ρ (h (rho)has units. ρ is caed resistivity. (R. Boton - 202) Resistance (R. Boton - 202) Resistance 2 Physics 55.3: Introduction to Eectricity and Magnetism Physics 55.3: Introduction to Eectricity and Magnetism 2
4 Resistivities (Conductors) Materia ρ at 20 C (Ω m) Auminum 2.82x0-8 Ω m Copper.72x0-8 Ω m God 2.44x0-8 Ω m Iron 9.70x0-8 Ω m Mercury 95.8x0-8 Ω m Nichrome (aoy) 00x0-8 Ω m Siver.59x0-8 Ω m Tungsten 5.60x0-8 Ω m Carbon 3500x0-8 Ω m Resistance The resistance of a ength of wire is given by: R = ρ A where: R is resistance. R has units Vot/Amperes. The ratio of V/A is given the name ohms and written Ω (capita Greek etter omega). (R. Boton - 202) Resistance 3 (R. Boton - 202) Resistance 4 Physics 55.3: Introduction to Eectricity and Magnetism 3 Physics 55.3: Introduction to Eectricity and Magnetism 4 Exampe Copper Wire Resistance A copper wire is km ong and has a cross sectiona area of cm 2 (i.e., 22 AWG (American Wire Gauge)). What is the resistance of the wire? Note: The temperature of the wire is 20 C. Exampe Soution: (52.2Ω) Note: Diameter of 22 AWG wire is 0.64mm. House wiring (4 AWG) is.63mm (R. Boton - 202) Resistance 5 (R. Boton - 202) Resistance 6 Physics 55.3: Introduction to Eectricity and Magnetism 5 Physics 55.3: Introduction to Eectricity and Magnetism 6
5 Resistivity of an Insuator In an insuator, eectrons become free once in a whie. They get knocked free by a coision with a neighboring i atom (therma vibration) if that atom has enough energy. The eectron is then acceerated by the eectric fied to a neighboring atom, where it wi orbit for a ong time (making the atom a negative ion) unti it is knocked k free. Note that it wi take ess energy to free an eectron from a negative ion. (R. Boton - 202) Resistance 7 (R. Boton - 202) Resistance 8 Physics 55.3: Introduction to Eectricity and Magnetism 7 Physics 55.3: Introduction to Eectricity and Magnetism 8 Resistivities (Insuators) Ceary the average drift veocity is much smaer because of the time the free eectron is trapped by the negative ion (i.e., the time it spends in orbit of the negative ion). Therefore, the resistivity of insuators is much higher than that of conductors. Materia ρ at 20 C C (Ω m) Mica Rubber (hard) Tefon Wood (Mape) Ω m Ω m 0 6 Ω m 3x0 0 Ω m (R. Boton - 202) Resistance 9 (R. Boton - 202) Resistance 20 Physics 55.3: Introduction to Eectricity and Magnetism 9 Physics 55.3: Introduction to Eectricity and Magnetism 20
6 Resistivity of a Semiconductor Semiconductors are insuators when they are cod and conductors when they are hot. It does not take as much energy to knock an eectron free, so at room temperature there are ots of free eectrons at any one time. The outer eectrons of negative ions are not as tighty bound as in insuators, so the eectrons are not hed up as ong. (R. Boton - 202) Resistance 2 (R. Boton - 202) Resistance 22 Physics 55.3: Introduction to Eectricity and Magnetism 2 Physics 55.3: Introduction to Eectricity and Magnetism 22 Temperature Effects Increasing the temperature creates more free eectrons. In conductors there are aready so many free eectrons that this does not reay affect the resistivity. Moecuar vibration increases with temperature. t The vibrating atom occupies a arger space causing the time between coisions to get smaer. Reducing the time between coisions decreases the average drift veocity of the eectrons and increases resistivity. Resistivities (Semiconductors) Materia ρ at t20 C C( (Ω m) Carbon Germanium Siicon 3500x0-8 Ω m 0.50 Ω m 2000 Ω m (R. Boton - 202) Resistance 23 (R. Boton - 202) Resistance 24 Physics 55.3: Introduction to Eectricity and Magnetism 23 Physics 55.3: Introduction to Eectricity and Magnetism 24
7 Reca that R = ρ A Therefore if ρ increases with temperature, so does R. Therefore, the resistance of a conductor increases with temperature. (R. Boton - 202) Resistance 25 (R. Boton - 202) Resistance 26 Physics 55.3: Introduction to Eectricity and Magnetism 25 Physics 55.3: Introduction to Eectricity and Magnetism 26 Inferred Absoute Zero Using simiar trianges R R 2 T2 T0 = T T 0 T 2 T = R T T 0 R2 0 (R. Boton - 202) Resistance 27 (R. Boton - 202) Resistance 28 Physics 55.3: Introduction to Eectricity and Magnetism 27 Physics 55.3: Introduction to Eectricity and Magnetism 28
8 Inferred Absoute Zeros Materia T o Siver -243 C Copper CC Auminum -236 C Tungsten -204 CC (R. Boton - 202) Resistance 29 Exampe 2 Resistance at -35 C Consider a spoo of copper wire. The ength of the wire spoo is 000 m. The cross sectiona area of the wire is cm 2. What is the resistance of the wire at -35 C? Note: The resistance of the copper wire at 20 C is 52.2Ω (i.e., this is the same wire from the previous exampe): (R. Boton - 202) Resistance 30 Physics 55.3: Introduction to Eectricity and Magnetism 29 Physics 55.3: Introduction to Eectricity and Magnetism 30 Exampe 2 Soution: (40.85Ω) Resistance and Temperature It is possibe to transate the resistance of a wire into a temperature. Suppose you have a resistor made from a conductor and the resistance is known to be R at temperature T. The resistance is measured after the temperature of the resistor is changed and found to be R 2. The new temperature, T 2, is given by: (R. Boton - 202) Resistance 3 (R. Boton - 202) Resistance 32 Physics 55.3: Introduction to Eectricity and Magnetism 3 Physics 55.3: Introduction to Eectricity and Magnetism 32
9 T2 R2 = R T T 0 T 0 R2 [ T T0 ] = T2 T0 R where: T 2 is the temperature of resistor R 2, 2 2 T is the temperature of resistor R, T 0 is the inferred absoute zero temperature for the materia being used. T R2 [ T ] 2 = T0 + T0 R (R. Boton - 202) Resistance 33 (R. Boton - 202) Resistance 34 Physics 55.3: Introduction to Eectricity and Magnetism 33 Physics 55.3: Introduction to Eectricity and Magnetism 34 Exampe 3 Tungsten Resistor A tungsten resistor is used as a temperature sensor. The resistance of the tungsten resistor is Ω at 0 C. What is the temperature of the tungsten resistor, if its resistance is Ω? Note: T 0 for tungsten is C. Exampe 3 Soution: (20.4 C) (R. Boton - 202) Resistance 35 (R. Boton - 202) Resistance 36 Physics 55.3: Introduction to Eectricity and Magnetism 35 Physics 55.3: Introduction to Eectricity and Magnetism 36
10 Temperature Coefficient A second equation for cacuating the resistance of a resistor at temperature T is: R = R20[ + α20( T 20 C) ] where: R is the resistance at temperature T, R 20 is the resistance at 20 C (the subscript must be the same as the temperature in the brackets). α 20 is the temperature coefficient for the materia used in the resistor at a temperature of 20 C. Compare R = R [ + ( T 20 C) ] with 20 α 20 R2 R + T2 T T T0 ( ) = It is obvious that α20 = 20 T 0 (R. Boton - 202) Resistance 37 (R. Boton - 202) Resistance 38 Physics 55.3: Introduction to Eectricity and Magnetism 37 Physics 55.3: Introduction to Eectricity and Magnetism 38 Temperature Coefficients Materia α at 20 C ( C - ) Auminum 3.9x0-3 Copper 3.93x0-3 God 3.4x0-3 Iron 5.5x0-3 Siver 3.8x0-3 Tungsten 5.0x0-3 Carbon -0.5x0-3 Siicon -0.7x0-3 In both semiconductors and insuators, the number of free eectrons is a strong function (and aso a compicated function) of temperature. The number of free eectrons increases with temperature, t giving i rise to a arger current if the eectric fied is the same. Therefore, resistance decreases with temperature unti the materia gets very hot and reativey few free eectrons are gained by increasing temperature. (R. Boton - 202) Resistance 39 (R. Boton - 202) Resistance 40 Physics 55.3: Introduction to Eectricity and Magnetism 39 Physics 55.3: Introduction to Eectricity and Magnetism 40
11 Photo Resistors The point of minimum resistance depends on the materia. Photo resistors are made from a semiconductor such as siicon. The incident ight creates free eectrons which decreases the resistivity. The brighter the ight incident on the photo resistor, the ower the resistance. These devices can be used to turn off street ights during the dayight hours. (R. Boton - 202) Resistance 4 (R. Boton - 202) Resistance 42 Physics 55.3: Introduction to Eectricity and Magnetism 4 Physics 55.3: Introduction to Eectricity and Magnetism 42
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