Electric Currents. Resistors (Chapters 2728)


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1 Electric Currents. Resistors (Chapters 2728) Electric current I Resistance R and resistors Relation between current and resistance: Ohm s Law Resistivity ρ Energy dissipated by current. Electric power Electromotive force: emf, ε Simple resistive circuits: Series and parallel circuits. Circuits reducible to simple combinations Circuits nonreducible to series and parallel: Kirchhoff Rules Resistors and capacitors combined: dc RC circuit
2 Electric Current Charge Carrier Motion in a Conductor In an electrically insulated wire, the electrons undergo thermal random motion When an potential difference is set up between the ends of a wire, the surface charge gets nonuniformly distributed, producing an electric field inside the wire, such that electrons tend to drift against the average field producing an electric current, I more positive V E E a E c a b b + E c b + E + net E + net When a circuit is completed, the electric field sets in the wires with the speed of light (of the order of 10 8 m/s), and all free electrons throughout the material move very fast; however, their motion is impeded by collisions in the material, so their drift speed, v d, is much smaller, (of the order of 10 4 m/s) Ex: The zigzag lines represent the drift against the electric field of an electron in a conductor. The drift is analog with the parabolic trajectories of a ball drifting due to gravity down a wall with pegs. The nature of the charges carrying the current depends on the nature of the material Ex: electrons in metals, electrons and ions in plasma, holes in semiconductors, etc. e ion cores E v d more negative V m g pegs
3 Electric Current Definition So, whenever electric charges of like signs drift, an electric current is said to exist In order to define it quantitatively, consider at the charge flowing through a wire perpendicular to a crosssectional surface of area A Def: The rate at which the electric charge flows through this surface, that is, the amount of charge dq flowing per unit of time dt through the surface, is called an electric current dq 1C I SI Ampere (A) 1s dt Comments: Conventional current direction: the direction of the current is the direction of the drift of positive charge: that is, in the direction of the average field, or from high potential to low potential Albeit its directionality, the current is not a vector The current across a potential difference is the same through any crosssection, that is, the carriers drift identically as long as the conductive substance is the same The colloquial short for Amperes is Amps I Order of magnitude: flashlight bulb ~1A, sensitive electronics ~μa, highpower devices (such as large electromagnets) ~ka Q A
4 Electric Current Relation to drift speed. Current density To find the relationship between the macroscopic electric current and the microscopic details of the electriccarrier drift, consider elementary charge carriers of charge e drifting with constant drift speed v d through a current carrying conductor of crosssectional area A, with the charge concentration given by: Number of carriers n Volume Then, as shown on the figure, Qe n Q neax Ax Therefore, combining with the definition of current, we find that the current is related to the drift speed v d as following: Q x I nea I neav t t d So, if the drift speed is small (many collisions), the current is small and vice versa. If the carriers are electrons (charge e), the current opposes the drift velocity. Def: The current per unit crosssectional area of a conductor is a vector called current density: J I A nq v d
5 Quiz 1: We learned that the electron drift speed is relatively small: of the order of 10 4 m/s. How come, when the circuit in the figure is closed, the bulb light up almost instantaneously? a) Actually it doesn t: in my house we wait minutes until bulbs light up b) Since the energy of the electric field is delivered to the bulb before electrons from the battery reach to it c) Because the sea of electrons fill all the circuit. When the circuit is closed, the field is set with the speed of light and electrons start to drift through each crosssection of the circuit including the bulb. Quiz 2: Both segments of the shown wire are made of the same metal. Current I 1 flows into segment 1 from the left. How does current I 1 in segment 1 compare to current I 2 in segment 2? a) I 1 > I 2 b) I 1 = I 2 c) I 1 < I 2 d) There s not enough information to compare them
6 Ohm s Law Statement Experiments show that in most metals the current density J depends on: the electric field E the tabulated properties of the material: resistivity ρ or conductivity σ = ρ 1 The dependency is given by Ohm s Law: In an ohmic (or linear) material at constant temperature, the resistivity and conductivity remain constant for any electric field and the current density is proportional to the field: Comments: J E E J constant Ohm s Law does not state the proportionality E ~ J, but the constancy of ρ For most metals, resistivity increases approximately linearly with temperature T, over limited Tranges: 1 T T 0 0 ρ is the resistivity at temperature T ρ 0 is the resistivity at a reference temperature T 0 (usually taken to be 20 C) is the temperature coefficient of resistivity Ex: Only some metals follow this behavior through the whole range of temperatures. The resistivity of superconductors drops sharply to zero under a critical T, while the resistivity of semiconductors decreases with increasing T ρ ρ ρ ρ 0 Slope = αρ 0 T 0 conductor T T c superconductor T T seminconductor
7 Resistance and Resistors A new element of circuit Ohm s law can be reformulated in terms of the potential difference V ab responsible for the electric field driving a current through a wire: V I L V a L A A A I ab E J Vab I Vab RI length parallel with current 1V The quantity R is termed the resistance of the conductor: R SI Ohm (Ω) 1A The elements of circuit with resistance are called resistors. Symbol: a I V V V IR V V ab a b a b R +  b E J area perpendicular on current In a resistor in circuit with a battery, the electric potential decreases in the direction of the current the resistor determines a voltage drop while the potential increases across a battery it determines a voltage raise since a battery does work on the carriers I I I Analogy: circuit of pipes with water in laminar flow L Constricted pipe resistance Flow rate current High Low V b Pump: pressure difference potential
8 Quiz 3: An ohmic material is probed for drawn current I by applying an increasing potential difference V across the sample. Which of the shown I vs. V graphs (called IVcharacteristics) represents the likely dependency of I on V? I I a) b) V V Quiz 4: What geometrical aspect of the IVcharacteristics is a measure of the resistance? a) The intercept of V axis divided by the respective current b) The slope of the graph c) The inverse of the slope of the graph Problems: 1. Temperature dependence of current: A resistor with uniform crosssection and temperature coefficient α = C 1 is heated from T 0 = 20 o C to T = 170 o C. All this time a constant potential difference is applied across the device. If the current through the resistor at T 0 is I 0 = 2.0 A what is the current at temperature T? 2. Resistance and directionality: A metallic solid parallelepiped with resistivity ρ has length L, width a and height b. Find the resistance of the object if a) a potential difference is applied on the ab crosssections b) the potential difference is applied perpendicular on L
9 Meters in a Circuit Measuring current and voltage An ammeter is used to measure current. Symbol: A It must be mounted in line with the element along which the current is measured: all the charge passing through the element must also pass through the meter In order to measure current without modifying it, the ammeter must have a very small resistance, so no potential difference across it A voltmeter is used to measure potential difference. Symbol: It must be connected to the two ends of the element In order to measure the voltage without affecting it, the voltmeter must carry no current so its resistance is very large V
10 Energy Dissipated in a Resistor Qualitative approach We ve seen that the capacitor is an element of circuit that stores electric energy. What about a resistor? What does a resistor do from an energy point of view? Consider an element of positive charge, dq, moving around a closed circuit from point a back to point a As the charge moves through the battery from a to b, the potential energy of the charge increases by Vdq (on behalf of the chemical energy of the battery) As the charge moves through the resistor R, from c to d, it looses energy in collisions with the atoms of the resistor: the energy is transferred to the internal energy ground: taken as having zero potential When the charge returns to a, the net result is that some chemical energy of the battery has been delivered to the resistor and caused its temperature to rise: we say that the energy was dissipated across the resistor Notice that as long the charge flowing around the circuit (that is, the current) looses energy in the resistor, the battery must resupply it in order to maintain the current Resistance acts like friction in a mechanics: without resistance (friction), the current would flow (charges would move) around the circuit without the need of a battery
11 Say that a circuit carries a charge dq through a difference of potential V across a resistor R that dissipates energy du = Vdq The rate at which the energy U is dissipated is the electric power of the resistor: du dq P V P IV dt dt Using Ohm s law, we can find two useful alternative forms: The unit of energy used by electric companies is the kilowatthour: defined in terms of the unit of power and the amount of time it is supplied: 1 kwh = J Quiz: Energy Dissipated in a Resistor Electric power 5. Which of the resistors on the right will draw a larger current? a) A b) B c) Both draw the same current P 2 2 V I R R 6. Which of the two resistors will get hotter? a) A b) B c) Both will get equally hot d) They don t get hot
12 Types of Circuits DirectCurrent or dc circuits are traveled by currents in only one direction: the magnitude of the currents along the circuit branches may vary, but not their direction AlternatingCurrent circuits or ac circuits are traveled by currents in directions alternating with a certain periodicity Different elements of circuit behave differently in dc and ac circuits Electromotive Force Sources and internal resistance Def: An device such as a battery or generator that maintains the current in a closed circuit is said to provide an electromotive force or emf, ε Volts V SI A real battery has some internal resistance r, so some of its energy is dissipated internally Therefore, the voltage V ab across a real battery when it drives a current is not equal to the emf V V Ir ab The voltage V ab is applied to the external circuit of resistance R (called load), so r IR Ir The internal resistance can be represented as a resistor in series with the battery
13 Electromotive Force Comments Notice that the emf ε is equal to the terminal voltage when the current is zero (also called the opencircuit voltage) If the emf source is mounted across a load, a current I is driven and the difference of potential across the battery decreases by Ir The current provided by the battery depends both on the load R and the internal resistance r I R r When R r, the source is considered as ideal Ordinary batteries have a small internal resistance, such that the voltage delivered is less than the nominal voltage which is the emf The battery power is distributed both to the load and internal resistance: So, when R r, most of the power delivered by the battery is transferred to the load 2 2 I I R I r ε r IR Ir Ex: Voltage diagram The voltage V raised by the battery by ε drops both across the internal resistance r and the load R R
14 V V V s In general, for n resistors in series: s ab bc IR IR IR s 1 2 R R R 1 2 R R R R R Electric Circuits Resistors in series When two or more resistors are connected endtoend, they are said to be in series The current is the same in all resistors because any charge that flows through one resistor flows through the other: I I I 1 2 As a consequence of energy conservation around the circuit, the sum of the potential differences across the resistors is equal to the total potential difference across the combination n The series equivalent resistance is always greater than any of the individual resistors V ab V bc R s
15 Electric Circuits Resistors in parallel When two or more resistors are connected to the same two points in the circuit, they are said to be in parallel The potential difference across each resistor is the same As a consequence of charge conservation, the current, that enters into a junction must be equal to the total current leaving that junction, so, using for instance junction a, I I I 1 2 V V V Rp R1 R RR 1 2 Rp R R R R R p V V V In general, for n resistors in parallel: R R R R p 1 2 V V V 1 ab V 2 ab So, the parallel equivalent resistance is always less than the smallest resistor n R p
16 Problems: 3. Current with and without internal resistance: A battery with emf ε and internal resistance r = 2.0 Ω delivers a current I 0 = 100 ma when connected to a load R = 70 Ω. Calculate the current I through the circuit if the internal resistance of the battery were zero. 4. Analysis of combined circuit: An electric circuit contains an ideal battery with ε and four given resistors, R 1, R 2, R 3, R 4, arranged as in the figure. a) Calculate the equivalent resistance of the circuit. b) Calculate the current through resistor R 4 and the potential difference V bc across it c) Calculate the potential difference V ab across points a and b on the circuit. d) Calculate the currents I 1 and I 2 through R 1 and R 2, 3 respectively. e) Say that point c is grounded. What are the potentials in points a, b, and d with respect to this ground? What changes if the ground changes? a ε R 1 R 2 d R 3 c R 4 b
17 Kirchhoff s Rules Statements Resistors can be connected so that the circuits formed cannot be reduced to a single equivalent resistor as a combination of parallel and series arrangements These generic circuits can be analyzed that is, can be described in terms of currents carried along various branches, and potential differences between different points in the circuit using Kirchhoff s Rules: 1. Junction Rule A statement of Charge Conservation The sum of the currents entering any junction must equal the sum of the currents leaving that junction I 0 2. Loop Rule A statement of Energy Conservation The sum of the potential differences across all the elements around any closed circuit loop must be zero V 0
18 Kirchhoff s Rules Junction rule I in I out How to use the rule: Assign symbols and directions to the currents in all branches of the circuit The directions can be arbitrary: if a direction is chosen incorrectly, the current resulting after solving the equations will be negative, with a correct magnitude Write the junction rule for as many junctions as needed as needed, as long as each time you write an equation you include in it a current that has not been used in a previous junction rule equation In general, the number of times the junction rule can be used is one fewer than the number of junction points in the circuit Ex: Hydrodynamic analog: the current entering into a junction splits into partial currents (a) as water flowing into bifurcating pipes (b)
19 Kirchhoff s Rules Loop rule How to use the rule: Vi around a loop Circuits contain loops of electric elements such as resistors and batteries Choose an arbitrary direction to travel around a circuit loop Resistors: 0 Ex: clockwise loop If the resistor is traveled in the direction of the current: V = IR (voltage drop) If the resistor is traveled opposite to the current: V = +IR (voltage raise) Batteries: If the source of emf is traveled in the direction of the emf: V = +ε (voltage raise) If the source of emf is traveled opposite to the emf: V = ε (voltage drop)
20 Kirchhoff s Rules Problem solving strategy 1. Draw the circuit diagram and assign labels and symbols to all quantities 2. Assign directions to the currents no need for the directions to be all correct; a current with an incorrectly chosen direction will eventually come out negative 3. Apply the junction rule to any junction in the circuit 4. Apply the loop rule to as many loops as are needed to solve for the unknowns 5. Solve the equations simultaneously for the unknown quantities. Check your answers Problems: a 5. Applying Kirchhoff s Rules one battery: An electric circuit contains an ideal battery with ε = 6 V and four given resistors, R 1 = 10 Ω, R 2 = 12 Ω, R 3 = 2 Ω, arranged as in the figure. Use Kirchhoff Rules to calculate the currents I, I 1, I 2 through the branches of the circuit. ε b R 1 R 2 6. Applying Kirchhoff s Rules two batteries: An electric circuit contains 4 resistors and two ideal batteries, as in the figure. Write out Kirchhoff rules for this circuit. R 4 a ε 1 R2 ε 2 c R 1 d b R 3
21 RC Circuit Functionality A more complex behavior is expected when resistors are connected in the same circuit with capacitors forming an RC dccircuit: in these circuits, currents will be unidirectional (dc), but their magnitudes will vary with time When the RC circuit is completed is series with a battery ε, the capacitor C starts to charge and the circuit is called in charging regime Due to the presence of resistance R, the flow of charge is slowed down, such that the capacitor will take time to approach its maximum charge Q = Cε The capacitor builds a potential opposing the battery, so the current i which is initially ε/r when there is no charge on the capacitor decreases until the capacitor is fully charged and the current in the circuit tends to zero Subsequently, if the battery is removed, the capacitor will enter in a discharging regime gradually releasing its charge through the resistor like a finitecharge reservoir Initially the capacitor drives the current ε/r opposite to the current in the charging regime, but the current decreases since the charge on the capacitor depletes ready for charging ready for discharging
22 RC Circuit Charging regime Using Kirchhoff rules around the circuit, one ca obtain an equation for how the charge on the capacitor increases with time exponentially, tending to Q final = Cε: q Q 1e final t RC time constant τ = RC charging Consequently, the current through resistor decreases with time from I 0 = ε/r to zero: i I e 0 t RC The constant = RC is called the time constant and represents the time required for the charge to increase from zero to 63.2% of its maximum In a circuit with a large time constant, the capacitor charges and discharges slowly
23 RC Circuit Discharging regime When the battery is removed, the capacitor discharges and it can be shown that the charge decreases exponentially with time, from Q 0 asymptotically to zero: discharging q t Q0e time constant τ The current decreases exponentially from I 0 (but in opposite direction than when charging) i t I0e I 0 V R Q RC 0 0 maximum voltage across the capacitor
24 Problems: 7. Charging an RC circuit: Demonstrate that the time dependency of the charge on the plates of a capacitor in an RCcircuit charged by a battery with emf ε is, indeed, given by 1 t RC t RC 1 f q C e Q e 8. RC circuit: An RC circuit is connected to a battery with emf ε = 10 V. The capacitance is C = 0.50 μf and the resistance is R = 4.0 MΩ. a) Calculate the current and the amount of charge accumulated in the capacitor at t = 0.20 s in the charging regime b) How long will it take until the charge reaches half of its maximum value c) If the battery is removed (discharging regime), how long will it take until the charge in the capacitor is half the maximum charge d) What is the current through the resistor at that moment?
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