Electric current and direct-current circuits Chapter 21

Electric current Section 21-1

Electric current Electric current – A flow of electric charge from one place to another. Often, the charge is carried by electrons moving through a metal wire (much like water molecules flowing through a hose or blood cells flowing through an artery). Definition of electric current, I 𝐼= ∆𝑞 ∆𝑡 Where ∆𝑞 is a charge and ∆𝑡 is time SI unit: coulomb per second, C/s = ampere, A ”Amp” is short for ampere: 1 A = 1 C/s

Turn it up! The disk drive in a portable CD player is connected to a battery that supplies it with a current of 0.22 A. How many electrons pass through the drive in 4.5 s?

Electric potential Definition: Electric potential energy per charge Represented using V SI unit: joule/coloumb = volt, V (the variable that represents electric potential and the variable that represents its unit are the same: V) In common usage, the electric potential is often referred to simply as the “potential”.

Circuits When charge flows through a closed path and returns to its starting point, we refer to the closed path as an electric circuit. We will deal only with direct current circuits, also known as DC circuits, in which the current always flows in the same direction.

Batteries Although electrons move rather freely in metal wires, they do not flow unless the wires are connected to a source of energy. (water hose analogy) Battery – Uses chemical reactions to produce a difference in electric potential between its two ends, or terminals. The terminal corresponding to a high electric potential is denoted by a +, and the terminal corresponding to a low electric potential is denoted by a -. When the battery is connected to a circuit, electrons move in a closed path from the negative terminal of the battery, through the circuit, and back to the positive terminal.

Batteries A simple example of an electrical system is a battery, a switch, and a light bulb as they might be connected in a flashlight. In the schematic circuit shown, the switch is “open” – creating an OPEN CIRCUIT – which means there is no closed path through which the electrons can flow. Would the light be on or off?

Electromotive forces When a battery is disconnected from a circuit and carries no current, the difference in electric potential between its terminals is referred to as its ELECTROMOTIVE FORCE, or emf (𝜀). It follows that the units of emf are the same as those of electric potential: volts Clearly it’s not a force at all; instead, the emf determines the amount of work a battery does to move a certain amount of charge around a circuit. 𝑊=∆𝑞𝜀 PRACTICE!! A battery with an emf of 1.5 V delivers a current of 0.44 A to a flashlight bulb for 64 s. Find (a) the charge that passes through the circuit and (b) the work done by the battery.

Real life The emf of a battery is the potential difference it can produce between its terminals under ideal conditions. In real batteries, however, there is always some internal loss, leading to a potential difference that is less than the ideal value. The greater the current flowing through a battery, the greater the reduction in potential difference between its terminals (just like friction works!) BECAUSE MOST BATTERIES HAVE RELATIVELY SMALL INTERNAL LOSSES, WE SHALL TREAT BATTERIES AS IDEAL – ALWAYS PRODUCING A POTENTIAL DIFFERENCE EQUAL TO 𝜀 - UNLESS SPECIFICALLY STATED OTHERWISE.

SCHEMATICS When we draw an electric circuit, it will be useful to draw an arrow indicating the flow of current. The direction of the current in an electric circuit is the direction in which a positive test charge would move. In typical circuits the charges that flow are actually negatively charged electrons. As a result, the flow of electrons and the current arrow point in opposite directions.

Resistance and Ohm’s law Section 21-2

Resistance Electrons flow through metal wires with relative ease. Ideally, nothing about the wire would prevent their free motion. Real wires, however, under normal conditions, always affect the electrons to some extent, creating a resistance to their motion in much the same way that friction slows a box sliding across the floor. Resistance: Represented by R Measured in Ohms, Ω 1 Ohm is 1 Volt/Amp

OHM’s law Ohm’s Law V = IR SI unit: volt, V Materials that follow Ohm’s law are said to be ohmic in their behavior; they show a simple linear relationship between the voltage applied to them and the current that results. (It would show a straight line with a constant slope of 1/R on a graph.) Nonohmic materials, have more complex relationships between voltage and current. (A plot of current versus voltage for nonohmic material is nonlinear - the material does not have a constant resistance.) Nonohmic materials are good for use in electronic devices, including LEDs.

Practice A potential difference of 24 V is applied to a 150-Ω resistor. How much current flows through the resistor? Schematic symbol for resistor

Resistivity Resistivity – The quantity that characterizes the resistance of a given material. Represented by 𝜌, measured in units of Ω∙𝑚 The resistivity of a wire not only depends on the material, but also length and cross- sectional area (water hose example). Definition of resistivity 𝑅= 𝜌𝐿 𝐴 Where L is length of the wire, A is its cross- sectional area, and R is the resistance Wire 1 has a length L and a circular cross section of diameter D. Wire 2 is constructed from the same material as wire 1 and has the same shape but its length is 2L, and its diameter is 2D. Is the resistance of wire 2 (a) the same as that of wire 1, (b) twice that of wire 1, or (c) half that or wire 1?

A current-carrying wire A current of 1.82 A flows through a copper wire 1.75 m long and 1.10 mm in diameter. Find the potential difference between the ends of the wire. (The value of 𝜌 for copper may be found in Table 21-1.)

Temperature Dependence and Superconductivity Wires carrying electric current can get quite hot. Wire filaments in an incandescent light bulb can get up to 2800ºC (the sun’s surface is about 5500ºC) As a wire is heated, its resistivity tends to increase. Superconductors are materials that when the temperature is cooled to extreme temperatures, the resistance drops to zero! The temperature at which this happens is different for each superconductor is is called the critical temperature. (We now know that this is the result of quantum effects – we won’t study this further but its very interesting!) https://www.ted.com/talks/boaz_almog_levitate s_a_superconductor?language=e (start at 4:25)

Energy and Power in Electric circuits Section 21-3

Electrical Power P = IV SI unit: watt, W Thus, 1 amp flowing across a potential difference of 1 V produces a power of 1 W. A handheld electric fan operates on a 3.00-V battery. If the power generated by the fan is 2.24 W, what is the current supplied by the battery?

MORE practice A battery that produces a potential difference V is connected to a 5-W light bulb. Later, the 5-W light bulb is replaced with a 10-W light bulb. (a) In which case does the battery supply more current? (b) Which light bulb has the greater resistance? A battery with an emf of 12 V is connected to a 545-Ω resistor. How much energy is dissipated in the resistor in 65 s?

Your Goose is cooked When you get a bill from the local electric company, you will find the number of kilowatt-hours of electricity that you have used. Notice that kilowatt-hour (kWh) has the units of energy (1 kWh = 3.6 x 106 J). Thus the electric company is charging for the amount of energy you use – and not for the rate at which you use it. (show electric bill) A holiday goose is cooked in the kitchen oven for 4.0 h. Assume that the stove draws a current of 20.0 A, operates at a voltage of 220.0 V, and uses electrical energy that costs $0.068 per kWh. How much does it cost to cook your goose?

Resistors in Series and parallel Section 21-4

Multiple resistors Electric circuits often contain a number of resistors connected in various ways. For each type of circuit that has multiple resistors, we calculate the EQUIVALENT RESISTANCE produced by a group of individual resistors for use in formulas. V = IReq

Resistors in series When resistors are connected one after the other, end to end, we say that they are in SERIES. The potential difference through each resistor will add up to be the value of the emf. The current is the same through each resistor since the are all on the same path. (Since batteries in real life have internal losses, the simplest way to model a real battery is to imagine it to consist of an ideal battery of emf in series with an internal resistance.) What does it look like to have something connected in series?? (demo) To find equivalent resistance of resistors in series: Req = R1 + R2 + R3 +… A circuit consists of three resistors connected in series to a 24.0-V battery. The current in the circuit is 0.0320 A. Given that R1 = 250.0 Ω and R2 = 150.0 Ω, find (a) the value of R3 and (b) the potential difference across each resistor.

Resistors in parallel Resistors are in PARALLEL when they are connected across the same potential difference and the current has parallel paths through which it can flow. The current through each resistor will add up to be the value of total current in the circuit. The potential difference in each resistor is the emf of the circuit. What does it look like to have something connected in parallel? (demo) To find equivalent resistance of resistors in parallel: 1 𝑅 𝑒𝑞 = 1 𝑅 1 + 1 𝑅 2 + 1 𝑅 3 +… Thus, the more resistors we connect in parallel, the smaller Req. Each time we add a new resistor in parallel, we give the battery a new path through which current can flow – analogous to opening an additional lane of traffic on a busy highway.

Three resistors in Parallel Short circuit Three resistors in Parallel If any one of the resistors in a parallel connection is equal to zero, Req is zero. This is a short circuit. In this case, all current flows through the path of zero resistance. Consider a circuit with three resistors, R1 = 250.0 Ω, R2 = 150.0 Ω, and R3 = 350.0 Ω, connected in parallel with a 24.0-V battery. Find (a) the total current supplied by the battery and (b) the current through each resistor.

Power dissipated Calculate the power dissipated in the two circuit problems we just analyzed (series and parallel). Which dissipates more power? Why? Two identical light bulbs are connected to a battery, either in series or in parallel. Are the bulbs in series (a) brighter than, (b) dimmer than, or (c) the same brightness as the bulbs in parallel? NOW LET’S TEST IT!

Combination circuits The rules we have developed for series and parallel resistors can be applied to more complex circuits. Try the given problem to the right! Find the equivalent resistance, then find the current of the circuit.

Combination special In the circuit shown in the diagram, the emf of the battery is 12.0 V, and each resistor has a resistance of 200.0 Ω. Find (a) the current supplied by the battery to this circuit and (b) the current through the lower two resistors.

Kirchoff’s rules Section 21-5

Kirchoff’s rules Kirchoff’s rules are simply ways of expressing charge conservation and energy conservation in a closed circuit. Kirchoff’s 2 rules: The junction rule The loop rule

The Junction rule Junction – Any point in a circuit where three or more wires meet. Kirchoff’s rule states: The algebraic sum of all currents meeting at any junction in a circuit must equal zero. We associate a + sign with currents entering a junction and a – sign with currents leaving a junction. If you don’t know the direction of the currents, simply choose a direction for the unknown currents, apply the junction rules, and continue as usual. If the value you obtain for a given current is negative, it simply means that the direction we chose was wrong; it actually flows in the opposite direction.

Example Use Kirchoff’s rule to find I2 in the top junction if I3 = 4 A and I1 = 9 A.

The loop rule Keep in mind that potential difference changes as it goes across a resistor since it requires that energy to force the current through the resistor. Kirchoff’s rule states: The algebraic sum of all potential differences around any closed loop in a circuit is zero. This is because potential drops across resistors.

Now use Kirchoff’s rules and your knowledge of parallel and series circuits to solve the following: Two loops and Two batteries Find I1 and I2. Find the currents in the circuit shown.

Ammeters and voltmeters Section 21-8

Devices for measuring electrical quantities ammeter voltmeter Device for measuring currents in a circuit. Designed to measure the flow of current through a particular portion of a circuit. To measure this current, insert the ammeter into the circuit in such a way that all the current flowing from A to B must also flow through the meter (connect it in series between A and B). Device for measuring voltage in a circuit. Designed to measure the potential drop between any two points in a circuit. To measure this voltage, connect the voltmeter in parallel to the circuit at the appropriate points you are trying to measure the potential drop across.

More info More on a voltmeter Multimeter A real voltmeter always allows some current to flow through it, which means that the current flowing through the circuit is less than before the meter was connected. As a result, the measured voltage is altered from its ideal value. An ideal voltmeter would be one in which the resistance is infinite, so the current it draws from the circuit is negligible. Sometimes the functions of an ammeter, voltmeter, and ohmmeter are combined in a single device called a multimeter. Adjusting the settings on a multimeter allows a variety of circuit properties to be measured.