13.1: Electric Circuits:Batteries and Resistors
The name electric current is given to the phenomenon that occurs when an electric field moves down a wire at close to the speed of light. Voltage is the electrical energy density (energy divided by charge) and differences in this density (voltage) cause electric current. Resistance is the amount a device in the wire resists the flow of current by converting electrical energy into other forms of energy. A device, the resistor, could be a light bulb, transferring electrical energy into heat and light or an electric motor that converts electric energy into mechanical energy. The difference in energy density across a resistor or other electrical device is called voltage drop.
In electric circuits (closed loops of wire with resistors and constant voltage sources) energy must be conserved. It follows that changes in energy density, the algebraic sum of voltage drops and voltage sources, around any closed loop will equal zero.
In an electric junction there is more than one possible path for current to flow. For charge to be conserved at a junction the current into the junction must equal the current out of the junction.
Key Concepts
 Ohm’s Law \begin{align*}V=IR\end{align*} (Voltage drop equals current times resistance.)

 This is the main equation for electric circuits but it is often misused. In order to calculate the voltage drop across a light bulb use the formula: \begin{align*}V_{lightbulb} = I_{lightbulb}R_{lightbulb}\end{align*}. For the total current flowing out of the power source, you need the total resistance of the circuit and the total voltage: \begin{align*}V_{total} = I_{total}R_{total}\end{align*}.
 Power is the rate that energy is released. The units for power are Watts \begin{align*}(W)\end{align*}, which equal Joules per second \begin{align*}[W] = [J]/[s]\end{align*}. Therefore, a \begin{align*}60\;\mathrm{W}\end{align*} light bulb releases \begin{align*}60\end{align*} Joules of energy every second.
The equations used to calculate the power dissipated in a circuit is \begin{align*}P=IV\end{align*}. As with Ohm’s Law, one must be careful not to mix apples with oranges. If you want the power of the entire circuit, then you multiply the total voltage of the power source by the total current coming out of the power source. If you want the power dissipated (i.e. released) by a light bulb, then you multiply the voltage drop across the light bulb by the current going through that light bulb.
Name  Electrical Symbol  Units  Analogy 

Voltage \begin{align*}(V)\end{align*}  Volts \begin{align*}(V)\end{align*} 
A water dam with pipes coming out at different heights. The lower the pipe along the dam wall, the larger the water pressure, thus the higher the voltage. Examples: Battery, the plugs in your house, etc. 

Current (\begin{align*}I\end{align*}) 
Amps \begin{align*}(A)\end{align*} \begin{align*}A = \mathrm{C/s}\end{align*} 
A river of water. Objects connected in series are all on the same river, thus receive the same current. Objects connected in parallel make the main river branch into smaller rivers. These guys all have different currents. Examples: Whatever you plug into your wall sockets draws current 

Resistance (\begin{align*}R\end{align*})  Ohm \begin{align*}(\Omega)\end{align*} 
If current is analogous to a river, then resistance is the amount of rocks in the river. The bigger the resistance the less current that flows Examples: Light bulb, Toaster, etc. 
 Resistors in Series: All resistors are connected end to end. There is only one river, so they all receive the same current. But since there is a voltage drop across each resistor, they may all have different voltages across them. The more resistors in series the more rocks in the river, so the less current that flows.
\begin{align*}R_{total} = R_1 + R_2 + R_3 + \ldots\end{align*}
 Resistors in Parallel: All resistors are connected together at both ends. There are many rivers (i.e. The main river branches off into many other rivers), so all resistors receive different amounts of current. But since they are all connected to the same point at both ends they all receive the same voltage.
\begin{align*}\frac{1} {R_{total}} = \frac{1} {R_1} + \frac{1}{R_2} + \frac{1} {R_3} + \ldots\end{align*}
 DC Power: Voltage and current flow in one direction. Examples are batteries and the power supplies we use in class. AC Power: Voltage and current flow in alternate directions. In the US they reverse direction 60 times a second. (This is a more efficient way to transport electricity and electrical devices do not care which way it flows as long as current is flowing. Note: your TV and computer screen are actually flickering 60 times a second due to the alternating current that comes out of household plugs. Our eyesight does not work this fast, so we never notice it. However, if you film a TV or computer screen the effect is observable due to the mismatched frame rates of the camera and TV screen.) Electrical current coming out of your plug is an example.
 Ammeter: A device that measures electric current. You must break the circuit to measure the current. Ammeters have very low resistance; therefore you must wire them in series.
 Voltmeter: A device that measures voltage. In order to measure a voltage difference between two points, place the probes down on the wires for the two points. Do not break the circuit. Volt meters have very high resistance; therefore you must wire them in parallel.
 Voltage source: A power source that produces fixed voltage regardless of what is hooked up to it. A battery is a reallife voltage source. A battery can be thought of as a perfect voltage source with a small resistor (called internal resistance) in series. The electric energy density produced by the chemistry of the battery is called emf, but the amount of voltage available from the battery is called terminal voltage. The terminal voltage equals the emf minus the voltage drop across the internal resistance (current of the external circuit times the internal resistance.)
Key Equations
\begin{align*}I = \triangle q/ \triangle t\end{align*}
current is the rate at which charge passes by; the units of current are Amperes \begin{align*}(1\;\mathrm{A} = 1\;\mathrm{C/s})\end{align*}.
\begin{align*}\triangle V = I \cdot R\end{align*}
the current flow through a resistor depends on the applied electric potential difference across it; the units of resistance are Ohms \begin{align*}(1 \Omega = 1\;\mathrm{V/A})\end{align*}.
\begin{align*}P = I \cdot \triangle V\end{align*}
the power dissipated by a resistor is the product of the current through the resistor and the applied electric potential difference across it; the units of power are Watts \begin{align*}(1\;\mathrm{W} = 1\;\mathrm{J/s})\end{align*}.
Electric Circuits Problem Set
 The current in a wire is 4.5 A.
 How many coulombs per second are going through the wire?
 How many electrons per second are going through the wire?
 A light bulb with resistance of \begin{align*}80\ \Omega\end{align*} is connected to a \begin{align*}9\;\mathrm{V}\end{align*} battery.
 What is the electric current going through it?
 What is the power (i.e. wattage) dissipated in this light bulb with the \begin{align*}9\;\mathrm{V}\end{align*} battery?
 How many electrons leave the battery every hour?
 How many Joules of energy leave the battery every hour?
 A \begin{align*}120\;\mathrm{V}, 75\;\mathrm{W}\end{align*} light bulb is shining in your room and you ask yourself…
 What is the resistance of the light bulb?
 How bright would it shine with a \begin{align*}9\;\mathrm{V}\end{align*} battery (i.e. what is its power output)?
 A bird is standing on an electric transmission line carrying \begin{align*}3000\;\mathrm{A}\end{align*} of current. A wire like this has about \begin{align*}3.0 \times 10^{5}\ \Omega\end{align*} of resistance per meter. The bird’s feet are \begin{align*}6\;\mathrm{cm}\end{align*} apart. The bird, itself, has a resistance of about \begin{align*}4 \times 10^5\ \Omega.\end{align*}
 What voltage does the bird feel?
 What current goes through the bird?
 What is the power dissipated by the bird?
 By how many Joules of energy does the bird heat up every hour?
 Which light bulb will shine brighter? Which light bulb will shine for a longer amount of time? Draw the schematic diagram for both situations. Note that the objects on the right are batteries, not resistors.
 Regarding the circuit to the right.
 If the ammeter reads \begin{align*}2\;\mathrm{A}\end{align*}, what is the voltage?
 How many watts is the power supply supplying?
 How many watts are dissipated in each resistor?
 Three \begin{align*}82\ \Omega\end{align*} resistors and one \begin{align*}12\ \Omega\end{align*} resistor are wired in parallel with a \begin{align*}9\;\mathrm{V}\end{align*} battery.
 Draw the schematic diagram.
 What is the total resistance of the circuit?
 What will the ammeter read for the circuit shown to the right?
 Draw the schematic of the following circuit.
 What does the ammeter read and which resistor is dissipating the most power?
 Analyze the circuit below.
 Find the current going out of the power supply
 How many Joules per second of energy is the power supply giving out?
 Find the current going through the \begin{align*}75\ \Omega\end{align*} light bulb.
 Find the current going through the \begin{align*}50\ \Omega\end{align*} light bulbs (hint: it’s the same, why?).
 Order the light bulbs in terms of brightness
 If they were all wired in parallel, order them in terms of brightness.
 Find the total current output by the power supply and the power dissipated by the \begin{align*}20\ \Omega\end{align*} resistor.
 You have a \begin{align*}600\;\mathrm{V}\end{align*} power source, two \begin{align*}10\ \Omega\end{align*} toasters that both run on \begin{align*}100\;\mathrm{V}\end{align*} and a \begin{align*}25\ \Omega\end{align*} resistor.
 Show me how you would wire them up so the toasters run properly.
 What is the power dissipated by the toasters?
 Where would you put the fuses to make sure the toasters don’t draw more than 15 Amps?
 Where would you put a \begin{align*}25\end{align*} Amp fuse to prevent a fire (if too much current flows through the wires they will heat up and possibly cause a fire)?
 Look at the following scheme of four identical light bulbs connected as shown. Answer the questions below giving a justification for your answer:
 Which of the four light bulbs is the brightest?
 Which light bulbs are the dimmest?
 Tell in the following cases which other light bulbs go out if:
 bulb \begin{align*}A\end{align*} goes out
 bulb \begin{align*}B\end{align*} goes out
 bulb \begin{align*}D\end{align*} goes out
 Tell in the following cases which other light bulbs get dimmer, and which get brighter if:
 bulb \begin{align*}B\end{align*} goes out
 bulb \begin{align*}D\end{align*} goes out
 Refer to the circuit diagram below and answer the following questions.
 What is the resistance between \begin{align*}A\end{align*} and \begin{align*}B\end{align*}?
 What is the resistance between \begin{align*}C\end{align*} and \begin{align*}B\end{align*}?
 What is the resistance between \begin{align*}D\end{align*} and \begin{align*}E\end{align*}?
 What is the the total equivalent resistance of the circuit?
 What is the current leaving the battery?
 What is the voltage drop across the \begin{align*}12\ \Omega\end{align*} resistor?
 What is the voltage drop between \begin{align*}D\end{align*} and \begin{align*}E\end{align*}?
 What is the voltage drop between \begin{align*}A\end{align*} and \begin{align*}B\end{align*}?
 What is the current through the \begin{align*}25\ \Omega\end{align*} resistor?
 What is the total energy dissipated in the \begin{align*}25\ \Omega\end{align*} if it is in use for 11 hours?
 In the circuit shown here, the battery produces an emf of \begin{align*}1.5\;\mathrm{V}\end{align*} and has an internal resistance of \begin{align*}0.5\ \Omega\end{align*}.
 Find the total resistance of the external circuit.
 Find the current drawn from the battery.
 Determine the terminal voltage of the battery
 Show the proper connection of an ammeter and a voltmeter that could measure voltage across and current through the \begin{align*}2\ \Omega\end{align*} resistor. What measurements would these instruments read?
 Students measure an unknown resistor and list their results in the Table (below); based on their results, complete the following:
 Show a circuit diagram with the connections to the power supply, ammeter and voltmeter.
 Graph voltage vs. current; find the bestfit straight line.
 Use this line to determine the resistance.
 How confident can you be of the results?
 Use the graph to determine the current if the voltage were \begin{align*}13\;\mathrm{V}\end{align*}.
Voltage \begin{align*}(v)\end{align*}  Current \begin{align*}(a)\end{align*} 

\begin{align*} 15\end{align*}  \begin{align*}.11\end{align*} 
\begin{align*}12\end{align*}  \begin{align*}.08\end{align*} 
\begin{align*}10\end{align*}  \begin{align*}.068\end{align*} 
\begin{align*}8\end{align*}  \begin{align*}.052\end{align*} 
\begin{align*}6\end{align*}  \begin{align*}.04\end{align*} 
\begin{align*}4\end{align*}  \begin{align*}.025\end{align*} 
\begin{align*}2\end{align*}  \begin{align*}.01\end{align*} 
 Students are now measuring the terminal voltage of a battery hooked up to an external circuit. They change the external circuit four times and develop the Table (below); using this data, complete the following:
 Graph this data, with the voltage on the vertical axis.
 Use the graph to determine the emf of the battery.
 Use the graph to determine the internal resistance of the battery.
 What voltage would the battery read if it were not hooked up to an external circuit?
Terminal Voltage \begin{align*}(v)\end{align*}  Current \begin{align*}(a)\end{align*} 

\begin{align*}14.63\end{align*}  \begin{align*}.15\end{align*} 
\begin{align*}14.13\end{align*}  \begin{align*}.35\end{align*} 
\begin{align*}13.62\end{align*}  \begin{align*}.55\end{align*} 
\begin{align*}12.88\end{align*}  \begin{align*}.85\end{align*} 
 Students are using a variable power supply to quickly increase the voltage across a resistor. They measure the current and the time the power supply is on. Use the Table (below) that they developed to complete the following:
 Graph voltage vs. current
 Explain the probable cause of the anomalous data after \begin{align*}8\end{align*} seconds
 Determine the likely value of the resistor and explain how you used the data to support this determination.
 Graph power vs. time
 Determine the total energy dissipation during the \begin{align*}18\end{align*} seconds.
Time(sec)  Voltage \begin{align*}(v)\end{align*}  Current \begin{align*}(a)\end{align*} 

\begin{align*}2\end{align*}  \begin{align*}10\end{align*}  \begin{align*}1.0\end{align*} 
\begin{align*}4\end{align*}  \begin{align*}20\end{align*}  \begin{align*}2.0\end{align*} 
\begin{align*}6\end{align*}  \begin{align*}30\end{align*}  \begin{align*}3.0\end{align*} 
\begin{align*}8\end{align*}  \begin{align*}40\end{align*}  \begin{align*}3.6\end{align*} 
\begin{align*}10\end{align*}  \begin{align*}50\end{align*}  \begin{align*}3.8\end{align*} 
\begin{align*}12\end{align*}  \begin{align*}60\end{align*}  \begin{align*}3.5\end{align*} 
\begin{align*}14\end{align*}  \begin{align*}70\end{align*}  \begin{align*}3.1\end{align*} 
\begin{align*}16\end{align*}  \begin{align*}80\end{align*}  \begin{align*} 2.7\end{align*} 
\begin{align*}18\end{align*}  \begin{align*} 90\end{align*}  \begin{align*}2.0\end{align*} 
 You are given the following three devices and a power supply of exactly \begin{align*}120\;\mathrm{v}\end{align*}. \begin{align*}^*\text{Device}\ X\end{align*} is rated at \begin{align*}60\;\mathrm{V}\end{align*} and \begin{align*} 0.5\;\mathrm{A}\end{align*} \begin{align*}^*\text{Device}\ Y\end{align*} is rated at \begin{align*}15\;\mathrm{w}\end{align*} and \begin{align*}0.5\;\mathrm{A}\end{align*} \begin{align*}^*\text{Device}\ Z\end{align*} is rated at \begin{align*}120\;\mathrm{V}\end{align*} and \begin{align*}1800\;\mathrm{w}\end{align*} Design a circuit that obeys the following rules: you may only use the power supply given, one sample of each device, and an extra, single resistor of any value (you choose). Also, each device must be run at their rated values.
 Given three resistors, \begin{align*}200\ \Omega, 300\ \Omega\end{align*} and \begin{align*}600\ \Omega\end{align*} and a \begin{align*}120\;\mathrm{V}\end{align*} power source connect them in a way to heat a container of water as rapidly as possible.
 Show the circuit diagram
 How many joules of heat are developed after 5 minutes?
 Construct a circuit using the following devices: a \begin{align*}120\;\mathrm{V}\end{align*} power source. Two \begin{align*}9\ \Omega\end{align*} resistors, device A rated at \begin{align*}1\;\mathrm{A}\end{align*}, \begin{align*}6\;\mathrm{V}\end{align*}; device \begin{align*}B\end{align*} rated at \begin{align*}2\;\mathrm{A}\end{align*}, \begin{align*}60\;\mathrm{V}\end{align*}; device \begin{align*}C\end{align*} rated at \begin{align*}225\;\mathrm{w}\end{align*}, \begin{align*}3\;\mathrm{A}\end{align*}; device \begin{align*}D\end{align*} rated at \begin{align*}15\;\mathrm{w}\end{align*}, \begin{align*}15\;\mathrm{V}\end{align*}.
 You have a battery with an emf of \begin{align*}12\;\mathrm{V}\end{align*} and an internal resistance of \begin{align*}1.00\ \Omega\end{align*}. Some \begin{align*}2.00\;\mathrm{A}\end{align*} are drawn from the external circuit.
 What is the terminal voltage
 The external circuit consists of device \begin{align*}X\end{align*}, \begin{align*}0.5\;\mathrm{A}\end{align*} and \begin{align*}6\;\mathrm{V}\end{align*}; device \begin{align*}Y\end{align*}, \begin{align*}0.5\;\mathrm{A}\end{align*} and \begin{align*}10\;\mathrm{V}\end{align*}, and two resistors. Show how this circuit is connected.
 Determine the value of the two resistors.
 Students use a variable power supply an ammeter and three voltmeters to measure the voltage drops across three unknown resistors. The power supply is slowly cranked up; use the data Table (below) to complete the following:
 Draw a circuit diagram, showing the ammeter and voltmeter connections.
 Graph the above data with voltage on the vertical axis.
 Use the slope of the bestfit straight line to determine the values of the three resistors.
 Quantitatively discuss the confidence you have in the results
 What experimental errors are most likely might have contributed to any inaccuracies.
Current \begin{align*}(ma)\end{align*}  Voltage \begin{align*}R_1\;\mathrm{(v)}\end{align*}  Voltage \begin{align*}R_2\;\mathrm{(v)}\end{align*}  Voltage \begin{align*}R_3\;\mathrm{(v)}\end{align*} 

\begin{align*}100\end{align*}  \begin{align*} 2.1\end{align*}  \begin{align*}3.6\end{align*}  \begin{align*}5.1\end{align*} 
\begin{align*}150\end{align*}  \begin{align*}3.0\end{align*}  \begin{align*}5.0\end{align*}  \begin{align*}7.7\end{align*} 
\begin{align*}200\end{align*}  \begin{align*}3.9\end{align*}  \begin{align*}7.1\end{align*}  \begin{align*}10.0\end{align*} 
\begin{align*}250\end{align*}  \begin{align*}5.0\end{align*}  \begin{align*}8.9\end{align*}  \begin{align*}12.7\end{align*} 
\begin{align*}300\end{align*}  \begin{align*}6.2\end{align*}  \begin{align*}10.8\end{align*}  \begin{align*}15.0\end{align*} 
\begin{align*}350\end{align*}  \begin{align*}7.1\end{align*}  \begin{align*}12.7\end{align*}  \begin{align*}18.0\end{align*} 
\begin{align*}400\end{align*}  \begin{align*}7.9\end{align*}  \begin{align*}14.3\end{align*}  \begin{align*}20.0\end{align*} 
\begin{align*}450\end{align*}  \begin{align*}9.0\end{align*}  \begin{align*}16.0\end{align*}  \begin{align*}22.0\end{align*} 
\begin{align*}500\end{align*}  \begin{align*}10.2\end{align*}  \begin{align*}18.0\end{align*}  \begin{align*}25.0\end{align*} 
\begin{align*}600\end{align*}  \begin{align*}12.5\end{align*}  \begin{align*}21.0\end{align*}  \begin{align*}31.0\end{align*} 
\begin{align*}700\end{align*}  \begin{align*}14.0\end{align*}  \begin{align*}25.0\end{align*}  \begin{align*}36.0\end{align*} 
Answers to Selected Problems

 \begin{align*}4.5\mathrm{C}\end{align*}
 \begin{align*}2.8 \times 10^{19}\end{align*} electrons
 \begin{align*}0.11 \;\mathrm{A}\end{align*}
 \begin{align*}1.0 \;\mathrm{W}\end{align*}
 \begin{align*}2.5 \times 10^{21}\end{align*} electrons
 \begin{align*}3636 \;\mathrm{W}\end{align*}
 \begin{align*}192 \ \Omega\end{align*}
 \begin{align*}0.42 \;\mathrm{W}\end{align*}
 \begin{align*}5.4 \;\mathrm{mV}\end{align*}
 \begin{align*}1.4 \times 10^{8} \;\mathrm{A}\end{align*}
 \begin{align*}7.3 \times 10^{11} \;\mathrm{W}\end{align*}, not a lot
 \begin{align*}2.6 \times 10^{7} \;\mathrm{J}\end{align*}
 left = brighter, right = longer
 \begin{align*}224 \;\mathrm{V}\end{align*}
 \begin{align*}448 \;\mathrm{W}\end{align*}
 \begin{align*}400 \;\mathrm{W}\end{align*} by \begin{align*}100 \ \Omega\end{align*} and \begin{align*}48 \;\mathrm{W}\end{align*} by \begin{align*}12 \ \Omega\end{align*}
 b. \begin{align*}8.3 \;\mathrm{W}\end{align*}
 \begin{align*}0.5\mathrm{A}\end{align*}
 .

\begin{align*}0.8\mathrm{A}\end{align*} and the \begin{align*}50 \ \Omega\end{align*} on the left
 \begin{align*}0.94 \;\mathrm{A}\end{align*}
 \begin{align*}112 \;\mathrm{W}\end{align*}
 \begin{align*}0.35 \;\mathrm{A}\end{align*}
 \begin{align*}0.94 \;\mathrm{A}\end{align*}
 \begin{align*}50, 45, 75 \ \Omega\end{align*}
 both \begin{align*}50 \ \Omega\end{align*} resistors are brightest, then \begin{align*}45 \ \Omega\end{align*}, then \begin{align*}75 \ \Omega\end{align*}
 \begin{align*}0.76 \;\mathrm{A}\end{align*}
 \begin{align*}7.0 \;\mathrm{W}\end{align*}
 b. \begin{align*}1000 \;\mathrm{W}\end{align*}
 .
 \begin{align*}9.1 \ \Omega\end{align*}
 \begin{align*}29.1 \ \Omega\end{align*}
 \begin{align*}10.8 \ \Omega\end{align*}
 \begin{align*} 26.8 \ \Omega\end{align*}
 \begin{align*}1.8\mathrm{A}\end{align*}
 \begin{align*}21.5\mathrm{V}\end{align*}
 \begin{align*}19.4\mathrm{V}\end{align*}
 \begin{align*}6.1\mathrm{V}\end{align*}
 \begin{align*}0.24\mathrm{A}\end{align*}
 \begin{align*}16 \;\mathrm{kW}\end{align*}
 \begin{align*}3.66 \ \Omega\end{align*}
 \begin{align*}0.36\mathrm{A}\end{align*}
 \begin{align*}1.32 \;\mathrm{V}\end{align*}
 .
 .
 .
 .
 .
 .
 \begin{align*}10\mathrm{V}\end{align*}
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