What will we learn in this chapter?

Transcription

1 Chapter 19: Current, resistance, circuits What will we learn in this chapter? What are currents? Resistance and Ohm s law (no, there are no 3 laws). Circuits and electric power. Resistors in series and parallel. Kirchhoff s rules. Resistor-Capacitor circuits. G. Kirchhoff ( )

2 Current So far: Focus on static fields ( electrostatics ) due to charges at rest. In an electrostatic situation there is no field inside a conductor. Now: We study systems where the fields inside materials cause charge motion. Simple definition: An electric current is any motion of charge from one region of a conductor to another. To keep the charges moving, we need a field E inside the conductor which produces a force F = qe. Definition: Current Think of a liquid: What is the amount of particles flowing trough an area A at a given time t? Here: What is the net charge Q flowing trough an area in time t. A Definition of current: When a net charge Q passes trough a cross section of a conductor during a time t, the current I is I = Q t Unit: 1 coulomb/second = 1 C/s = 1 A (Ampere). Note: The current is always a scalar quantity. Currents typically range from pa (computer) to 100 s A (car starter).

3 Current contd. A conventional current is treated as a flow of positive charges, regardless if the free charges in the conductor are positive or negative, or both. In a metallic conductor, the moving charges are electrons. Note that the current still points in the direction where positive charges would flow. Current contd. Closed loops: Due to conservation of charge, you can imagine a tube filled with peas. You push one in at one end, another one comes out on the other end. The current at any instant is the same at all cross sections. + + Notes: In circuit analysis we will always assume that currents consist of positive charges even though we know that for most conductors electrons are negatively charged.

4 Current contd. Electrons move around randomly in a metal. When you apply a potential (i.e., a field), a net drift is induced. Note that the E field travels at the speed of light in that medium. Analogy: Liquid In absence of gravity, molecules wiggle around. Adding a slope, the liquid flows. Resistance and Ohm s law Idea behind Ohm s law: The E-field causing a current in a conductor is associated to a potential difference. The current is proportional to the average drift speed of the moving charges, which is proportional to the E-field and thus to the potential. It follows, that the current I should be proportional to the potential V. I Mathematically: If I is proportional to V, then I = const V V

5 Resitance Definition of resistance: When the potential difference V between the ends of a conductor is proportional to the current I, the ratio V/I is called the resistance of the conductor: R = V I Unit: The SI unit is 1 ohm = 1 volt/ampere Ohm s law: The potential difference V between the ends of a conductor is proportional to the current I trough the conductor; the proportionality factor is the resistance R. Note: Ohm s law is only valid when V and I are proportional! This implies that R is a constant independent of V and I. Clearly, this is an idealized model. 1Ω = 1V/A Resistors Resistors are common parts in electric circuits. You will encounter them sooner or later in labs. Tables with the different color codings can be obtained on the internet. Example: Green:! 5 Violet:!! 7 Red:!! 10 2 Silver:!! 10% tolerance Result: 57x10 2 Ohm with 10% tolerance.

6 Resistor color codes Resistivity What happens with R if we double the diameter of a conductor? What happens with R if we double the length of a conductor? Can we define a geometry-independent quantity? Yes! Definition of resistivity: The resistance R is proportional to the length L and inversely proportional to the cross-section area A, with a proportionality factor ρ called the resistivity of the material: Note: The resistivity is material dependent and thus characterizes the conductance properties of given materials. Units: 1 ohm meter = 1Ωm R = ρ L A

7 Typical resistivities at room temperature Conductors have resistivities orders of magnitude smaller than insulators. These numbers depend on temperature Temperature dependence of resistance In general, the resistance of a metallic conductor increases with increasing temperature. Up to approximately 373K: R0 is a reference resistance at a given temperature T0 (usually 273 or 293K). For common metals α is between and K -1. Superconductors: R T = R 0 [1 + α(t T 0 )] The resistance drops to zero for temperatures less than Tc. Highest Tc known: 160K. Usually only a few K. Discovered 1911 by Kammerlingh- Onnes in Leiden.

8 Non-ohmic conductors I Ohmic conductor: The I V characteristic is linear with slope 1/R. Diode: Strongly nonlinear characteristic. Diodes act like one-way valves. Electromotive force (emf) What is needed such that a conductor has a steady current? A complete (closed) circuit. A potential difference across the circuit. A potential difference which overcomes the resistance of the path. How can we move charge from lower to higher potential such that we can later use it? emf: ElectroMotive Force E. Think of a water tower: We move water up there to later have pressure in the pipes. Note: It is not a force (units are 1 V = 1 J/C) but an energy.

9 Sources of emf Any device that converts energy of some form into electrical potential for a circuit. Examples: Batteries Solar cells Fuel cells Generators Nuclear fission ideal emf source not connected to a circuit The field associated with the potential difference Vab produces a force FE on the charges q. There is an intrinsic push (e.g., chemical reaction) in the device Fn moving charges from a to b that maintains the potential difference. When the emf source is not part of the circuit F n = F E and no charge flows.

10 ideal emf source connected to a circuit potential causes fields making charges move When an emf source is connected to a circuit with resistance R V ab = IR An electric field is set up in the wire and a current flows. Note: The current is the same at every point in the circuit (charge conservation ). Internal resitance in an emf source Ideal emf source: Charge inside the source does not loose energy when in motion. Real emf source: The movement of charge encounters resistance inside the source. If the internal resistance r follows Ohm s law, r is a constant. The current trough the emf source has an associated potential drop equal to Ir. The terminal potential difference is thus: V ab = E Ir = V Ir The terminal voltage Vab is thus smaller than the emf. It follows for a real emf source: I = E/(R + r) The internal resistance can be taken into account as a resistance in series.

11 Common circuit symbols Example: Source in an open circuit E = 12V r =2.0Ω What are the readings on the volt and ammeter? Voltmeter: There is no closed circuit. Hence no current flows. Since there is no current trough the battery, there is no potential difference across the internal resistance r. Thus V ab = E = 12V. Ammeter: There is no complete circuit. Hence no current flows and the ammeter reads 0A.

12 Example: Source in a closed circuit E = 12V R =4.0Ω r =2.0Ω What are the readings on the volt and ammeter? We have a closed circuit, thus a current flows: I = E r + R = Voltmeter: 12V 4Ω +2Ω = 2A V ab = IR = 2A 4Ω = 8V V ab = E Ie = 12V 2A 2Ω = 8V Example contd. The potential difference starts and ends at the same point. Note that the potential on the VM can be taken across the resistor or the battery itself. The potential between an old and new battery is the same. The internal resistance increases drastically with age.

13 Energy and power in electric circuits When a charge moves in a circuit, the electric field performs work on it. For a current I = Q/ t the associated charge does a work W = V ab Q = V ab I t The rate of work performed (energy transferred into the circuit) on a certain time is called power P. We obtain P = W = V ab I t Unit: 1 Watt = 1 W = (1 J/C)(1 C/s) = 1 J/s. If the potential at b is higher than a then Vab is negative and there is a net transfer of energy out of the circuit element. Pure resistance When a current flows trough a resistor, electric energy is transformed into thermal energy. Using Vab = IR it follows: P = V ab I = I 2 R = V 2 ab R What happens to this energy? Charges collide with atoms and increase the internal energy of the material. The energy is dissipated into the resistor. Overheating might cause damage. Conversely, we can use the heat (oven, toaster, hair dryer, ).

14 Power output of a (real) source The emf source converts energy into electrical energy at a rate EI. The internal resistance dissipates energy at a rate: P internal = I 2 r. Net power output: P = EI P internal = EI I 2 r Quiz: Why are LED lamps more efficient than incandescent bulbs? Think: which ones are hotter? Resistors in series and in parallel Why should we care? Vendors make a finite set of resistors, thus, if we want a given resistance for an appliance, we need to combine different ones. Two possible options: Series connection ( one after the other ) Parallel connection ( next to each other ) What is the equivalent resistance for a given circuit?

15 Resistors in series Characteristics: There is only one path for the current to flow and it has to go trough all resistors. It follows that the current I is the same in all resistors. Potentials across resistors: V ax = IR 1 V xy = IR 2 V yb = IR 3 V ab = V ax + V xy + V yb = I(R 1 + R 2 + R 3 ) R eq = R 1 + R 2 + R 3 Equivalent resistance for resistors in series: R eq = R 1 + R 2 + R 3 + R it is always the sum and greater than any individual resistance. Resistors in parallel Characteristics: There are different paths the current can take, but the potential across all resistors is the same. Currents across resistors: I 1 = V ab I 2 = V ab I 3 = V ab R 1 R 2 R 3 ( 1 Current conservation: I = I 1 + I 2 + I 3 = V ab ) R 1 R 2 R 3 Equivalent resistance for resistors in parallel: 1 = R eq R 1 R 2 R 3 R 4 it is always the reciprocal sum and smaller than any resistor.

16 Warning: Resistors vs Capacitors. The rules series vs parallel are inverted! Kirchhoff s rules Some networks cannot be reduced easily to parallel series combinations. Some examples: Battery-Charger-Lamp Rectifyer-like circuit Gustav Kirchhoff figured out simple rules to treat these difficult cases.

17 Kirchhoff s rules: some definitions Junction: a point where 3 or more conductors meet. Loop: a closed conducting path. Kirchhoff s rules Kirchhoff s junction (or point) rule:! The algebraic sum of the currents into any junction is zero; that is I =0 Kirchhoff s loop rule:! The algebraic sum of the potential differences in any loop, including! those associated with emf s and those of resistive elements, must! equal zero; that is V =0

18 Kirchhoff s rules contd. Notes: The junction rule is based on the conservation of electric charge (no charge can accumulate at a junction). Think again of a water pipe: If 2L/min flow in, then 2L/min flow out. The loop rule is based on the conservation of energy: If we walk along a path measuring potential differences, then the sum of all these when we reach the starting point has to be zero. 1L If this were not the case, the force would not be conservative! 2L 1L Example What are the potential differences across A, B, C when the switch is open and when it is closed? Note: I is conserved. Open: Only one loop. Since A, B, C are identical, it follows that the voltage is the same across all bulbs. Closed: Two loops. Since the switch is an ideal conductor, it follows from the small loop that VC = 0. The emf of the source is now split only between 2 bulbs and so the potential across A and B increases by 50%.

19 General problem solving strategy Draw a large circuit diagram so there is plenty of space to write within it. Often you will not know the actual current flow direction. No problem: Your result might simply need a sign switch. Whenever possible, apply the junction rule to remove as many unknown quantities as possible: General problem solving strategy contd. Choose any closed loop in the network, and designate a direction to go around it. Go around that loop in the designated direction adding potential differences as you cross them. Special considerations: An emf is counted positive if you cross it from to + and negative when crossed from + to. An IR product is negative if you path passes the resistor in the same direction as the assumed current and positive otherwise. Relative signs are essential!

20 General problem solving strategy contd. Apply Kirchhoff s loop rule to the previously-obtained potential differences. If necessary, choose a different loop to obtain another relation between the different unknowns until you have enough equations to determine them all. Solve the equations! Special case: calculation of a potential drop between points a and b Start at b and add the potential changes on the way to a. The algebraic sum of the potential changes is Vab = Va - Vb. Example: Charging a car battery What is the current I in the circuit? What is the potential difference Vab between points a and b? Note: there is only one loop, thus we do not need to use the junction rule.

21 Example contd. Set up: Assume the current flows from the + terminal of the charged battery. If this is not right, then the current will come out negative. Big deal Apply the loop rule: Start at a and use V = IR. I(4Ω) 4V I(7Ω) + 12V I(2Ω) I(3Ω) = 0 The in the resistor terms come from the fact that we have a potential drop. We find I = 0.5A. Example contd. Potential drop between a and b: Use the upper path from b to a. Remember that Vab = Va - Vb. Follow that loop! V b + 12V (0.5A)(2.0Ω) (0.5A)(3.0Ω) =V a Result Vab = 9.5V. Taking the lower path we obtain the same result.

22 Resistance Capacitance (RC) circuits So far we have assumed that resistances and emfs are constant. A simple example where this is not the case is an RC circuit. capacitor initially uncharged when the switch is closed the capacitor is charged, while the current decreases RC circuit contd. Initial setup: The capacitor is uncharged, i.e., the potential difference is zero and its initial charge Q0 = 0. Following Kirchhoff s rules, the voltage across R is equal to the terminal voltage E. The initial current I0 trough the resistor is I 0 = E/R (Ohm s). Switch closed: From Kirchhoff s loop rule it follows: E = ir + q C The solution of the differential equation is i = I 0 e t/rc q = Q final (1 e t/rc ) the relaxation time when i or q are 1/e is τ = RC. i = q/ t

23 RC circuit contd. Graphical representation of i and q: Strictly speaking, it takes an infinite amount of time for the current to be zero and the capacitor to be fully charged.