Direct Currents. We will now start to consider charges that are moving through a circuit, currents. Sunday, February 16, 2014


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1 Direct Currents We will now start to consider charges that are moving through a circuit, currents. 1
2 Direct Current Current usually consists of mobile electrons traveling in conducting materials Direct current (DC) is defined as a current that flows only in one direction in the conductor Computers, electronic devices are based on direct currents. Many of our electric technology is based on Alternating Current (AC) Chapter 26 13
3 Electric Current (1) We define the electric current i the charge passing a given crosssection in unit time A A Current = positive charge that passed through A to the right plus negative charge that passed through A to the left per unit time. i = A (q + n + v + +( q )n v ) 14
4 Direct Current We will study charges in motion Electric charge moving coherently from one region to another is called electric current. Do not need to move net charge to produce current, just relative motion of charges usually n + = n n + v + = n v electric wires are not charged! 13
5 Again, wires do not need to be charged to conduct current: A A n + = n i = ea (n + v + n v ) In a wire positive ions are at rest, v+ =0 i = ea (n v ) 5
6 Electric Current (2) The amount of charge q passing a given point in time t is the integral of the current with respect to time given by Charge conservation implies that charge flowing in a conductor is never lost Therefore in a steady DC current the same amount of charge must flow into one end of the conductor and exit from the other end of the conductor 15
7 The Ampere The unit of current is coulombs per second, which has been given the unit ampere, named after French physicist AndréMarie Ampère, ( ) The ampere is abbreviated as A and is defined by Some typical currents are Flashlight  1 A The starter motor in a car A ipod  50 ma Lightning strike (for a very short time) 100,000 A 16
8 Batteries We use batteries as devices that provide direct currents in circuits If you examine a battery, you will find its voltage written on it This voltage is the potential difference that it can provide to a circuit You will also find its rating in units of mah This rating provides information on the total charge that a single battery can deliver over its lifetime The quantity mah is another unit of charge: 17
9 Typical Battery electrochemical reactions: anions (negatively charged ions) migrate to anode, cations (positively charged ions) migrate to cathode. This creates potential difference 18
10 Current is a scalar Current Current has a sign but not a direction (Note: current density is a vector!  see later) We will represent the direction of the current flowing in a conductor using an arrow This arrow represents whether the net current is positive or negative in a conductor at a given point but does not represent a direction in three dimensions Physically, the charge carriers in a conductor are electrons that are negatively charged However, as is conventionally done, we define positive current as the net flow of positive charge carriers past a given point per unit time 19
11 Circuits In this circuit, electrons flow around the circuit counterclockwise. (The conventionally defined current is clockwise; remember, electrons are negative charges.) The electrons can t disappear so the DC current requires a whole loop! + lightbulb Chemical action pumps electrons from the positive terminal (+) to the negative terminal () in the battery. The emf (electromotive force, or electric field) pushes electrons around the wire from () to (+). 20
12 Current density: current flowing through unit area The current flowing through the surface is Current Density i = J da where is the differential area element perpendicular to the surface If the current is constant and perpendicular to the surface, then and we can write an expression for the magnitude of the current density 22
13 Current density: current flowing through unit area The current flowing through the surface is reminds you something? i = J da Current Density where is the differential area element perpendicular to the surface If the current is constant and perpendicular to the surface, then and we can write an expression for the magnitude of the current density 22
14 Current density: current flowing through unit area The current flowing through the surface is reminds you something? i = J da Current Density where is the differential area element perpendicular to the surface If the current is constant and perpendicular to the surface, then and we can write an expression for the magnitude of the current density Flux! 22
15 Drift Velocity (1) In a conductor that is not carrying current, the conduction electrons move randomly (thermal motion) When current flows through the conductor, the electrons have an additional coherent motion (drift velocity, v d ) The magnitude of the velocity of random thermal motion is on the order of 10 6 m/s while the magnitude of the drift velocity is on the order of 104 m/s We can relate the current density J to the drift velocity v d of the moving electrons j = e (n + v + n v ) Here v is averaged drift (!) velocity 23
16 Drift Velocity (4) Consider a wire carrying a current The physical current carriers are negatively charged electrons These electrons are moving to the left in this drawing However, the electric field, current density and current are directed to the right Comments Electrons are negative charges! On top of the coherent motion the electrons have random (thermal) motion. 26
17 Charge conservation: Gauss law for current Total charge inside a small volume can change only by flowing through its surface. Flow of charge = current! J j da = dq dt j j = dρ dt Law of charge conservation 15
18 Resistance and Resistivity Some materials conduct electricity better than others If we apply a given voltage across a conductor, we get a large current If we apply the same voltage across an insulator, we get very little current (ideal: none) The property of a material that describes its ability to conduct electric currents is called the resistivity, ρ The property of a particular device or object that describes its ability to conduct electric currents is called the resistance, R Resistivity is a property of the material; resistance is a property of a particular object made from that material 28
19 Resistance (1) If we apply an electric potential difference V across a conductor and measure the resulting current i in the conductor Experiment indicates that potential difference V and current i are proportional. Coefficient of proportionality: resistance R Ohm s law The unit of resistance is volt per ampere In honor of George Simon Ohm ( ) resistance has been given the unit ohm, Ω 29
20 Resistance (2) We will assume that the resistance of the device is uniform for all directions of the current; e.g., uniform metals The resistance, R, of a conductor depends on the material from which the conductor is constructed as well as the geometry of the conductor First we discuss the effects of the material and then we will discuss the effects of geometry on resistance 30
21 Resistivity The conducting properties of a material are characterized in terms of its resistivity We define the resistivity, ρ, of a material by the ratio E: magnitude of the applied field J: magnitude of the current density Another form of Ohm s law The units of resistivity are 31
22 Typical Resistivities The resistivities of some representative conductors at 20 C are listed in the table below (µωcm) As you can see, typical values for the resistivity of metals used in wires are on the order of 108 Ωm 32
23 Resistance Knowing the resistivity of the material, we can then calculate the resistance of a conductor given its geometry. Consider a homogeneous wire of length L and constant cross sectional area A. the resistance is and 33
24 Resistance and Resistivity For a wire, Resistance Resistivity 34
25 Example: Resistance of a Copper Wire (1) Standard wires that electricians put into residential housing have fairly low resistance. Question: What is the resistance of a length of 100 m of standard 12 gauge copper wire, typically used in household wiring for electrical outlets? Answer: The American Wire Gauge (AWG) size convention specifies wire cross sectional area on a logarithmic scale. A lower gauge number corresponds to a thicker wire. Every reduction by 3 gauges doubles the crosssectional area. 35
26 Example: Resistance of a Copper Wire (2) The formula to convert from the AWG size to the wire diameter is So a 12gauge copper wire has a diameter of 2.05 mm Its cross sectional area is then Look up the resistivity of copper in the table 36
27 Resistors In many electronics applications one needs a range of resistances in various parts of the circuits For this purpose one can use commercially available resistors Resistors are commonly made from carbon, inside a plastic cover with two wires sticking out at the two ends for electrical connection The value of the resistance is indicated by four colorbands on the plastic capsule The first two bands are numbers for the mantissa, the third is a power of ten, and the fourth is a tolerance for the range of values 37
28 Resistor Color Codes Example: Colors (left to right) red, yellow, green, and gold Using our table, we can see that the resistance is Ω = 2.4 MΩ with a tolerance of 5% 38
29 Temperature Dependence of Resistivity The resistivity (and hence resistance) varies with temperature For metals, this dependence on temperature is linear over a broad range of temperatures An empirical relationship for the temperature dependence of the resistivity of metals is given by Copper ρ is the resistivity at temperature T ρ 0 is the resistivity at some standard temperature T 0 α is the temperature coefficient of electric resistivity for the material under consideration 39
30 Temperature Dependence of Resistance In everyday applications we are interested in the temperature dependence of the resistance of various devices The resistance of a device depends on the length and the cross sectional area These quantities depend on temperature However, the temperature dependence of linear expansion is much smaller than the temperature dependence of resistivity of a particular conductor So the temperature dependence of the resistance of a conductor is, to a good approximation, 40
31 Temperature Dependence Our equations for temperature dependence deal with relative temperatures so that one can use ºC as well as K Values of α for representative metals are shown below 41
32 Origin of resistivity ions are at rest, electrons flow ions vibrate, electrons collide with ions and stop Stopping of electron: stopping of a current. At higher T, ions vibrate with larger amplitude: higher resistance electrons ions A 30
33 Other Temperature Dependence At very low temperatures the resistivity of some materials goes to exactly zero (not just small, exactly 0). These materials are called superconductors resistivity of a superconductor electrons do not collide with ions at all The resistance of some semiconducting materials actually decreases with increasing temperature number of conducting electrons depends on T 42
34 Power dissipated by circuit with resistively drifting electrons collide with ions and stop. energy of drift motion (energy of a flowing current) is given to ions. Ions are heated electrons ions A How much power a current dissipates? 32
35 Energy and Power in Electric Circuits (1) Consider a simple circuit in which a source of emf with voltage V causes a current i to flow in a circuit. The work required to move a differential amount of charge dq around the circuit is equal to the differential electric potential energy du given by So we can rewrite the differential electric potential energy as The definition of power P is Putting it together 65
36 Energy and Power in Electric Circuits (2) The power dissipated in a circuit or circuit element is given by the product of the current times the voltage Using Ohm s Law we can write equivalent formulations of the power with The unit of power is the watt (W) Electrical devices are rated by the amount of power they consume in watts Electricity bill is based on how many kilowatthours of electrical energy you consume kw h = power times time The energy is converted to heat, motion, light, 1 kw h = 1000 W X 3600 s = 3.6 x 10 6 joules 66
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