CK-12 People's Physics Book Version 2
Page 15
Chapter 15: Electric Circuits Version 2
The Big Ideas
In the last chapter, we looked at static configurations of charges. In general, problems with moving charges are very difficult to solve; the field that deals with these is called electrodynamics. In this chapter, we consider how charge can flow through conducting wires connecting opposite ends of a battery. Such a setup, called a circuit usually involves a current, a voltage source, and resistors.
Conductors have an effectively infinite supply of charge, so when they are placed in an electric field, a separation of charge occurs. A battery with a potential drop across the ends creates such an electric field; when the ends are connected with a wire, charge will flow across it. The term given to the flow of charge is electric current, and it is measured in Amperes (A) --- Coulombs per second. Current is analogous to a river of water, but instead of water flowing, charge does.
Voltage is the electrical energy density (energy divided by charge) and differences in this density (voltage) cause electric current. Batteries often provide a voltage difference across the ends of a circuit, but other voltage sources exist. If current is a river, differences in voltage can be thought of as pipes coming out of a water dam at different heights. The lower the pipe along the dam wall, the larger the water pressure, thus the higher the voltage.
Resistance is the amount a device in the wire resists the flow of current by converting electrical energy into other forms of energy. A 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. Resistance is analogous to rocks and other objects that impede the flow of water, transforming the water's kinetic energy into heat, sound, and other forms of energy through contact forces.
This is what a typical circuit looks like:
Circuit Basics
We use the following symbols to represent the quantities discussed above:
Name Symbol Electrical Symbol Units Everyday device
Voltage Volts (V) Battery, the plugs in your house, etc.
Current (flow of charge)
Amps (A)
Whatever you plug into your wall sockets draws current
Resistance Ohm Light bulb, Toaster, etc.
Loop and Junction Rules for Voltage/Current
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 or node 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.
Ohm's Law
The resistance of an object --- described above --- is quantified as the ratio of the voltage drop across it to the amount of current that will flow from that voltage. Note that the current depends on the voltage drop; here, as above we use instead of to mean voltage difference (both are accepted ways). Generally, more current flowing through a resistor will cause a higher voltage drop. For the special class of resistors discussed in this class this ratio is a constant --- the current flowing across these resistors will rise at the same rate as the voltage difference supplied. In other words, the resistance does not depend on the amount of current that flows through the resistor, or the voltage drop across it. This relationship is known as Ohm's Law, for a constant current it is usually written as Unlike equation [1], where varied with current, we can use equation [2] to find the current, voltage drop, or resistance across a resistor when given the other two. When dealing with a constant current, use equation [2], but when dealing with a battery driven circuit (a source of constant voltage difference), use equation [3].
Power
Power is the rate at which energy is lost by a system. The units of power are Watts (W), which equal Joules per second(1W = 1J/s). Therefore, a 60 W light bulb releases 60 Joules of energy every second.
The equation used to calculate the power dissipated in a circuit or across a resistor is: As with OhmÕs 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.
Resistors in Series and in Parallel
Sometimes, circuits have many resistors in various geometrical arrangements. When in series, two or more resistors are connected end to end (See picture). In this case the resistors receive the same current, but since they can have different resistances they may have different voltage drops across them. Analogously, there may be more rocks at some points in the river than in others, but if there is only one way for the river to flow, the current has to be the same at all points. It follows from Ohm's law that Since the total resistance will increase with each resistor added in series, adding resistors in series will cause the less current to flow at a set voltage (according to Ohm's Law for constant voltage sources, [3]).
When two or more resistors are connected together at both ends, they are said to be "in parallel" (see picture). There are many rivers (the river splits into streams), so all resistors receive different amounts of current. But since they all connect the same points on the circuit, the voltage drops across them have to be equal. The rule for combining resistors in parallel is Since the total resistance will decrease with the number or resistors in parallel, adding resistors in parallel to existing ones will cause more current to flow through a circuit.
Ohm's Law and Total Quantities
Ohm's law is the main relationship for electric circuits but it is often misused. In order to calculate the voltage drop across a light bulb --- or any single resistor --- use the formula: .
Using the formulas and the rules above, a circuit with any number of resistors (and voltage sources) can be modeled as a circuit with just one voltage source and one resistor, for which Ohm's Law also holds. For the total current flowing out of the power source, you need the total resistance of the circuit and the total current: This concept is illustrated below.
Example on Circuit Math
Question: Analyze the diagram below.
a) Find the current going out of the power supply.
b) How many Joules per second of energy is the power supply giving out?
c) Find the current going through the light bulb.
d) Order the light bulbs in terms of brightness.
e) If the light bulbs were all wired in parallel, order them in terms of brightness.
Answer
a) To find the current going out of the power supply, we will use equation [7], . We already have the total voltage drop and we are trying to solve for the current. This means that we need to know the total resistance before we can find the current.
To solve for the resistance we will apply the two rules for resistors (series and parallel) because we have both in are circuit. First, we must combine the two resistors in parallel so that we can treat the entire circuit as a series. According to equation [6], Because is equal to , we need to flip the fraction to get .
Now that we have three resistors in series (the two in parallel can be counted as one), we simply need to add them to get the total resistance.
We can now solve for the current by using equation [7] This is total net current through the circuit; it's also the current across the 50 resistors, but not the ones connected in parallel.
b) To find the power dissipated, we will use equation [4].
c) To find the current going through the light bulb, we must realize that a total of .94A goes through the two light bulbs in parallel; according to the junction rule above, the currents across the two light bulbs must add t
o this. Now we must find what proportion of the current the light bulb gets. To do this, we use our knowledge that resistors in parallel have the same voltage drop and Ohm's Law: d) The brightness is determined by the power dissipated. More power means a brighter lightbulb. According to equation [4], the power dissipated by a resistor can be written as . Since we know the resistance of and current across every resistor, we can simply calculate this quantity for each one. The order is 's, , then . The is brighter than the because the gets considerably more current.
e) When the bulbs are wired entirely in parallel, the voltage drops across them will be the same. Since , the way to determine the brightest bulb is to look at the currents across them, which will be inversely related with their resistances. So, the bulb with the lowest resistance will be the brightest, the one with the second lowest resistance will be second, and so on. Therefore the order is , , and finally .
Capacitors
When current flows through wires and resistors in a circuit as a result of a difference in electric potential, charge does not build up significantly anywhere on its path. Capacitors are devices placed in electric circuits where charge can build up. The amount of charge a capacitor can store before it “fills up” depends on its shape and how much electric potential is applied. The ratio of charge stored in a capacitor to the voltage applied is called its capacitance, measured in Farads . The larger the electric potential in volts, the stronger the electric field that is used to “cram” the charge into the device. Any capacitor will fill up with enough charge. Capacitors store energy when charged, and release it when they discharge.
Capacitors in Circuits (Steady-State)
When a capacitor is placed in a circuit, current does not actually travel across it. Rather, equal and opposite charge begins to build up on opposite sides of the capacitor --- mimicking a current --- until the electric field in the capacitor creates a potential difference across it that balances the voltage drop across any parallel resistors or the voltage source itself (if there are no resistors in parallel with the capacitor). The ratio of charge on a capacitor to potential difference across it is called capacitance:
Capacitors in Series and in Parallel
Like resistors, combinations of capacitors in circuits can be combined into one 'effective' capacitor. The rules for combining them are reversed from resistors:
Charging and Discharging Capacitors (Transient)
When a capacitor is initially uncharged, it is very easy to increase the amount of charge on its plates. As charge builds, the charge present repels new charge with more and more force. Due to this effect, the charging of a capacitor follows a logarithmic curve. When a circuit passes current through a resistor into a capacitor, the capacitor eventually Òfills upÓ and no more current flows across it. A typical RC circuit is shown below; when the switch is closed, the capacitor discharges with an exponentially decreasing current:
Charging a capacitor involves moving charges through a potential difference; as we saw in the electricity chapter, this results in electric potential energy being stored in the capacitor:
Capacitor Example
Question: Consider the figure above when switch S is open.
a) What is the voltage drop across the resistor?
b) What current flows through the resistor?
c) What is the voltage drop across the 20 microfarad capacitor?
d) What is the charge on the capacitor?
e} How much energy is stored in that capacitor?
Answer:
a) When the capacitor is charged --- in the steady state --- no current flows across it, and we basically have a circuit with two resistors in series. Accordingly, the voltage drop across the resistor will be in the same proportion to the net voltage across the circuit as its resistance is to the net resistance (see circuits chapter): This means that the voltage drop across the resistor is
b) Since there is only one path for the current to take, its value is the same everywhere on the circuit; all we have to do is find the total current. This will then also be the amount of current that flows through the resistor. We can find it by applying Ohm's Law for the circuit: Since we have the total resistance and the total voltage, we can solve for the total current using Ohm's law. The current flowing through the resistor is therefore .
c) We can find the voltage drop across the capacitor by realizing that the voltage drop across any parallel paths in a circuit have to be equal; otherwise the loop rule would be violated. Therefore, the voltage drop across the capacitor is the same as the voltage drop across the resistor. We can find this analogously to how we found the voltage drop across the other resistor:
d) To find the charge stored in the capacitor we will use the equation First we must convert the capacitor into the correct units for the equation. Then we can substitute in the values and solve for the charge stored.
e) The potential energy stored in a capacitor is All we need to do is plug in the known values and get the potential energy.
Key Terms
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 real-life 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.)
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 is connected to a battery. What is the electric current going through it?
What is the power (i.e. wattage) dissipated in this light bulb with the battery?
How many electrons leave the battery every hour?
How many Joules of energy leave the battery every hour?
A light bulb is shining in your room and you ask yourselfÉ What is the resistance of the light bulb?
How bright would it shine with a battery (i.e. what is its power output)?
A bird is standing on an electric transmission line carrying of current. A wire like this has about of resistance per meter. The birdÕs feet are apart. The bird, itself, has a resistance of about 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 resis
tors.
Regarding the circuit to the right. If the ammeter reads , what is the voltage?
How many watts is the power supply supplying?
How many watts are dissipated in each resistor?
Three resistors and one resistor are wired in parallel with a 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 light bulb.
Find the current going through the 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 resistor.
You have a power source, two toasters that both run on and a resistor. Show me how you would wire them up so the toasters run properly.