Electricity and Magnetism Current and Resistance Resistance and Resistivity

Transcription

1 Electricity and Magnetism Current and Resistance Resistance and Resistivity Lana Sheridan De Anza College Feb 5, 2018

2 Last time current current density drift velocity

3 : J (ne)v : d. Warm Up Question (26-7) unit The is the figure coulomb shows per conduction cubic meter electrons (C/m moving 3 ), is the leftward in a wire. itive Are carriers, the following ne is positive leftwardand or rightward: Eq predicts irection. (a) For the (conventional) negative carriers, current nei, is negative and J : s. (b) the current density J, (c) the electric field E in the wire? ectrons moving leftng leftward or righturrent density J, : (c) (A) all leftwards (B) all rightwards (C) leftward, leftward, rightward (D) rightward, rightward, leftward

4 : J (ne)v : d. Warm Up Question (26-7) unit The is the figure coulomb shows per conduction cubic meter electrons (C/m moving 3 ), is the leftward in a wire. itive Are carriers, the following ne is positive leftwardand or rightward: Eq predicts irection. (a) For the (conventional) negative carriers, current nei, is negative and J : s. (b) the current density J, (c) the electric field E in the wire? ectrons moving leftng leftward or righturrent density J, : (c) (A) all leftwards (B) all rightwards (C) leftward, leftward, rightward (D) rightward, rightward, leftward

5 Overview resistance resistivity conductivity Ohm s Law

6 form conductor of cross-sectional area A. Drift Speed Electrons with E = 0: a v d = The random motion of the charge carriers is modified by the field, and they have a drift velocity opposite the direction of the electric field. (J = I/A) in the x direction in a time interval motio charge carriers is modified by required for charge carriers repea the field, and they have a drift in th whose velocity magnitude opposite the is direction equal to the zigzag leng with the of an same the E-field: electric as that field. required for all is the app ch the circular area S at one end. With field this i E produ the DQ m5 oppo Dividing both sides of this equation You b conductor is intern ecule Ifrom avg 5 the a S v In reality, the speed d of the charg tempe drift b speed. To understand the mea Q uic I which the charge carriers are free e Figure thr is, the 27.3 potential (a) A schematic difference across it nae = J diagram reg motion ne of that the random is analogous motion of to the mot two charge carriers in a conductor repeatedly with the metal atoms, an in the absence of an electric field. The zigzagged drift velocity as is in zero. Figure (b) The 27.3a. As dis is applied across the conductor (for motion of the charge carriers in

7 Drift Speed of an Electron in Copper What is the drift speed of the conduction electrons in a copper wire with radius r = 900 µm when it has a uniform current I = 17 ma? Assume that each copper atom contributes one conduction electron to the current and that the current density is uniform across the wire s cross section.

8 Drift Speed of an Electron in Copper How many electrons per unit volume? Same as number of copper atoms: n = N A ρ M = ( mol 1 )( kg/m 3 ) kg/mol N A is Avagadro s number, M is the molar mass (kgs per mole of copper), and ρ is copper s density. n = m 3 This is the number of free conduction electrons in a cubic meter of copper. (A lot.)

9 Drift Speed of an Electron in Copper v d = = I nae ( A) ( m 3 )(πr 2 )( C)

10 Drift Speed of an Electron in Copper v d = = I nae ( A) ( m 3 )(πr 2 )( C) v d = m/s Very slow!

11 lectric field can s are available, Resistance ting loop is no ial making When a up potential difference is applied across a conductor, current (a) begins to flow. them to move electron flow i does not vary ucting loop in a hypothetical is defined as i i Battery + i However, different amounts of current will flow in different conductors, (26-1) even Fig. when 26-1 the applied (a) A loop potential of copper difference in is the same. electrostatic equilibrium. The entire time interval loop is at a single potential, and the What is the characteristic of the conductor which determines the electric field is zero at all points inside the amount of current that copper. will flow? (b) Adding a battery (26-2) imposes an electric potential difference between the ends of the loop (b) 1 Figure from Halliday, Resnick, Walker, 9th ed. i

12 Resistance Resistance The resistance of a conductor is given by the ratio of the applied potential to the current that flows through the conductor at that potential: R = V I The units of resistance are Ohms, Ω, symbol is the capital Greek letter Omega. 1 Ω = 1 V/A We can think of a high resistance as resisting, or impeding, the flow of current.

13 Resistivity An individual conductor or circuit component has a resistance. The resistance is based on the material it is made of, its geometry, and the temperature The material that a component is made from affects the resistance, because different materials have different resistivities.

14 Resistor A resistor is a component that can be incorporated into a circuit. It has a particular resistance at a given voltage. 1 Image from thomasnet.com

15 Resistor in a Circuit diagram

16 Resistivity resistivity, ρ the ratio of the electric field strength in a material to the current density this field causes in the material: ρ = E J Resistivity is a property of a material. Its symbol is the Greek letter ρ, pronounced rho. The units of resistivity are Ω m. 1 Ω m = 1 V A m = 1 V/m A/m 2 which agrees with the definition of ρ = E/J.

17 t diagram, we represent a resistor and ite Eq.26-8 as Resistivity, sistance,the smaller the current. on the manner in which the potential example, shows a given potential difthe same conductor. As the current the two cases hence the measured rwise stated, we shall assume that any ig. 26-8b. ifference to a conducting rod.the gray tance.when they are arranged as in ed resistance is larger than when they d. r connections, we often wish to take a jects but with materials. Here we do so across a particular resistor but on the ial. Instead of dealing with the current i : ent density J at the point in question. eal with the resistivity r of the material: finition of r). (26-10) according to Eq , we get, for the V Table 26-1 Resistivities of Some Materials at Room Temperature (20 C) Resistivity, r Temperature Material ( m) Coefficient of Resistivity, a (K 1 ) Typical Metals Silver Copper Gold Aluminum Manganin a Tungsten Iron Platinum Typical Semiconductors Silicon, pure Silicon, n-type b Silicon, p-type c Typical Insulators Glass Fused quartz 10 16

18 Resistivity Together with the geometry of the component made of that material, we can predict the resistance of the component.

19 Resistivity Together with the geometry of the component made of that material, we can predict the resistance of the component. For a wire, cylinder, or anything with uniform cross-section A, made of material with resistivity ρ: and L is the length of the wire. R = ρl A 690 CHAPTER 26 CURRENT AND RESISTANCE A i Current is driven by a potential difference. L V Fig A potential difference V is applied between the ends of a (This follows from the definition of ρ.) i We can write E Equations a electrical propertie We often speak cal of its resistivity,

20 an be applied only to a homogeneous isotropic conductor of ion, Question with the potential difference applied as in Fig. 26-8b. pic quantities V, i,and R are of greatest interest when we are measurements on specific conductors. They are the quantities Rank the three cylindrical copper conductors according to the tly on meters. We turn to the microscopic quantities E, J,and r current through them, greatest first, when the same potential ested in the fundamental electrical properties of materials. difference V is placed across their lengths. T 3 L shows three r conductors A 1.5L A_ face areas and 2 m according to h them, greatsame (a) (b) potential difference V is placed across their lengths. (A) a, b, c (B) c, b, a (C) b, (a and c) (D) (a and c), b A_ 2 L/2 (c)

21 an be applied only to a homogeneous isotropic conductor of ion, Question with the potential difference applied as in Fig. 26-8b. pic quantities V, i,and R are of greatest interest when we are measurements on specific conductors. They are the quantities Rank the three cylindrical copper conductors according to the tly on meters. We turn to the microscopic quantities E, J,and r current through them, greatest first, when the same potential ested in the fundamental electrical properties of materials. difference V is placed across their lengths. T 3 L shows three r conductors A 1.5L A_ face areas and 2 m according to h them, greatsame (a) (b) potential difference V is placed across their lengths. (A) a, b, c (B) c, b, a (C) b, (a and c) (D) (a and c), b A_ 2 L/2 (c)

22 Resistivity can depend on Temperature The reopper as a emperat on the a convence point at T ivity r 0 m. Resistivity (10 8 Ω. m) Room temperature (T 0, ρ 0 ) Temperature (K) Resistivity c on tempera with Temperature

23 Resistivity can depend on Temperature The relationship between resistivity and temperature is close to linear. For most engineering purposes, a linear model is good enough. The model: ρ ρ 0 = ρ 0 α(t T 0 ) The resistivity varies linearly with the difference in temperature from some reference value T 0. ρ 0 is the resistivity at T 0.

24 Resistivity can depend on Temperature ρ ρ 0 = ρ 0 α(t T 0 ) α is just a constant, however it takes different values for different materials. α is called the temperature coefficient of resistivity. It has units K 1. For example for copper 1 : ρ 0 = Ω m α = K 1 at 20 C 1 Value from Halliday, Resnick, Walker, 8th ed.

25 Conductivity Sometimes it is useful to represent how conductive a material is: how readily it permits current to flow, as opposed to how much it resists the flow of current. conductivity, σ a measure of what the current density is in a material for a particular electric field; the inverse of resistivity: σ = 1 ρ = J E

26 Conductivity Sometimes it is useful to represent how conductive a material is: how readily it permits current to flow, as opposed to how much it resists the flow of current. conductivity, σ a measure of what the current density is in a material for a particular electric field; the inverse of resistivity: σ = 1 ρ = J E This is different than surface charge density (also written σ). This is just an unfortunate coincidence of notation.

27 Conductivity conductivity, σ a measure of what the current density is in a material for a particular electric field; the inverse of resistivity: σ = 1 ρ = J E The units of conductivity are (Ω m) 1. We can use conductivity to relate the current density to the electric field in a material: J = σe

28 Resistance of Resistors with Non-Uniform Area For a resistor with uniform cross-section A, made of material with resistivity ρ: R = ρl A What if the cross section isn t uniform?

29 Resistance of Resistors with Non-Uniform Area For a resistor with uniform cross-section A, made of material with resistivity ρ: R = ρl A What if the cross section isn t uniform? Integrate. Use: dr = ρ A(l) dl A(l) means Area is a function of position, l, along the length of the conductor. (Not A times l.)

30 ential difference of 10 V. b lihe rs. The undesired as ecut ection is radial. Polyethylene tor stic are known, we th a of the plastic from sis- b he ed Inner Outer Find the resistance between conductor the two conducting conductor layers. alculate the resis- Example: Coaxial Cable f the problem. The sistance of a block ation. Because the Inner sition, conductor we must use a dr Outer conductor L a b Current direction dr r End view Current direction Figure 27.8 (Example 27.3) A

31 Example: Coaxial Cable Find the resistance between the two conducting layers. At radius r the area a current can pass through is A(r) = 2πrL R = b a ρ 2πrL dr

32 Example: Coaxial Cable Find the resistance between the two conducting layers. At radius r the area a current can pass through is A(r) = 2πrL R = = = b a ρ 2πrL dr ρ [ln b ln a] 2πL ρ 2πL ln ( ) b a

33 Ohm s Law Ohm s Law The current through a device is directly proportional to the potential difference applied across the device. V I Not all devices obey Ohm s Law! In fact, for all materials, if V is large enough, Ohm s law fails. They only obey Ohm s law when the resistance of the device is independent of the applied potential difference and its polarity (that is, which side is the higher potential).

34 V is high V Ohm s Law (fro +? As we just discussed in Section 26-4, a resisto (wit i resistance. Does not It Potential has obey that difference Ohm s same resistance (V) no matter cau law: (a) ( polarity) of the applied (b) potential difference ar ever, might have resistances that change with pas th Figure 26-11a shows how to distinguish is su t +2 V is applied across the device being tested, and dev +4 device is measured as V is varied in both magn diff 0 V is arbitrarily +2 taken to be positive when the dev higher potential than the right terminal. The 0 is m 2 (from left to right) is arbitrarily assigned a pl not (with the 2 right terminal at a higher potential Potential difference (V) causes is assigned Potential a minus difference sign. (V) Ohm (b) Figure 26-11b is (c) a plot of i versus V for one passing through the origin, so the ratio i/v (whi O We can write this linear relationship Fig. is the as26-11 V same = for (a) IRA all potential ifvalues and only difference of V. This RV means t is applied to the terminals of a device, pr is constant device is independent of the magnitude and +4 and independent of V establishing. a current i.(b) A plot of current i versus difference V. applied potential difference V (Th +2 when Figure the device 26-11c is a is 1000 a plot resistor. for another (c) A conducti term However, notice that we can always device define plot when only R( V the when device the ) = is a polarity V semiconducting I even of V is positive resi an when 0 resistance does depend on V is pnmore. junction than diode. about 1.5 V.When current does tion ex not linear; it depends on the value of the applied We distinguish between the two types ofa Potential difference (V) Ohm s Law Obeys Ohm s i law: Current (ma) Current (ma) Curre Current (ma)

35 CHECKPOINT 4 potential differen The following table gives the current i (in junction diode, we amperes) through two devices for several gives values the of potential current difference i (in amperes) V (in through pendent two of V. of Ohm s law, ho The following table devices for several volts). values From ofthese potential data, determine difference which V (in volts). We can expre Which of the devices obeys does not Ohm s obey Ohm s law? law. materials rather Eq (E : Device 1 Device 2 A conducting m V i V i independent of th Ohm s Law Question (A) 1 only (B) 2 only (C) both (D) neither Halliday, Resnick, Walker, page 692. All homogeneous ductors like pure within some rang there are departu

36 CHECKPOINT 4 potential differen The following table gives the current i (in junction diode, we amperes) through two devices for several gives values the of potential current difference i (in amperes) V (in through pendent two of V. of Ohm s law, ho The following table devices for several volts). values From ofthese potential data, determine difference which V (in volts). We can expre Which of the devices obeys does not Ohm s obey Ohm s law? law. materials rather Eq (E : Device 1 Device 2 A conducting m V i V i independent of th Ohm s Law Question (A) 1 only (B) 2 only (C) both (D) neither Halliday, Resnick, Walker, page 692. All homogeneous ductors like pure within some rang there are departu

37 Summary Resistance resistivity conductance Ohm s Law 2nd Test Thursday, Feb 15. Homework Collected homework 2, posted online, due on Monday, Feb 12. Serway & Jewett: PREVIOUS: Ch 27, onward from page 824. Problems: 1, 5, 7 NEW: Ch 27, Problems: 15, 23, 25, 29, 33, 71