# Electric Current, Resistance, and Ohm's Law

## Ohm’s Law: Resistance and Simple Circuits

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the origin of Ohm’s law.
2. Calculate voltages, currents, or resistances with Ohm’s law.
3. Explain what an ohmic material is.
4. Describe a simple circuit.
What drives current? We can think of various devices—such as batteries, generators, wall outlets, and so on—which are necessary to maintain a current. All such devices create a potential difference and are loosely referred to as voltage sources. When a voltage source is connected to a conductor, it applies a potential difference  that creates an electric field. The electric field in turn exerts force on charges, causing current.

### Ohm’s Law

The current that flows through most substances is directly proportional to the voltage  applied to it. The German physicist Georg Simon Ohm (1787–1854) was the first to demonstrate experimentally that the current in a metal wire is directly proportional to the voltage applied:
This important relationship is known as Ohm’s law. It can be viewed as a cause-and-effect relationship, with voltage the cause and current the effect. This is an empirical law like that for friction—an experimentally observed phenomenon. Such a linear relationship doesn’t always occur.

### Resistance and Simple Circuits

If voltage drives current, what impedes it? The electric property that impedes current (crudely similar to friction and air resistance) is called resistance . Collisions of moving charges with atoms and molecules in a substance transfer energy to the substance and limit current. Resistance is defined as inversely proportional to current, or
Thus, for example, current is cut in half if resistance doubles. Combining the relationships of current to voltage and current to resistance gives
This relationship is also called Ohm’s law. Ohm’s law in this form really defines resistance for certain materials. Ohm’s law (like Hooke’s law) is not universally valid. The many substances for which Ohm’s law holds are called ohmic. These include good conductors like copper and aluminum, and some poor conductors under certain circumstances. Ohmic materials have a resistance  that is independent of voltage  and current . An object that has simple resistance is called a resistor, even if its resistance is small. The unit for resistance is an ohm and is given the symbol  (upper case Greek omega). Rearranging  gives , and so the units of resistance are 1 ohm = 1 volt per ampere:
 shows the schematic for a simple circuit. A simple circuit has a single voltage source and a single resistor. The wires connecting the voltage source to the resistor can be assumed to have negligible resistance, or their resistance can be included in .
Resistances range over many orders of magnitude. Some ceramic insulators, such as those used to support power lines, have resistances of  or more. A dry person may have a hand-to-foot resistance of , whereas the resistance of the human heart is about . A meter-long piece of large-diameter copper wire may have a resistance of , and superconductors have no resistance at all (they are non-ohmic). Resistance is related to the shape of an object and the material of which it is composed, as will be seen in Resistance and Resistivity.
Additional insight is gained by solving  for  yielding
This expression for  can be interpreted as the voltage drop across a resistor produced by the flow of current . The phrase  drop is often used for this voltage. For instance, the headlight in  has an  drop of 12.0 V. If voltage is measured at various points in a circuit, it will be seen to increase at the voltage source and decrease at the resistor. Voltage is similar to fluid pressure. The voltage source is like a pump, creating a pressure difference, causing current—the flow of charge. The resistor is like a pipe that reduces pressure and limits flow because of its resistance. Conservation of energy has important consequences here. The voltage source supplies energy (causing an electric field and a current), and the resistor converts it to another form (such as thermal energy). In a simple circuit (one with a single simple resistor), the voltage supplied by the source equals the voltage drop across the resistor, since , and the same  flows through each. Thus the energy supplied by the voltage source and the energy converted by the resistor are equal. (See .)

### Test Prep for AP Courses

### Section Summary

1. A simple circuit is one in which there is a single voltage source and a single resistance.
2. One statement of Ohm’s law gives the relationship between current

  ,
voltage

  ,
and resistance

in a simple circuit to be

3. Resistance has units of ohms (

  ), related to volts and amperes by .
4. There is a voltage or  drop across a resistor, caused by the current flowing through it, given by .

### Conceptual Questions

### Problems & Exercises


# Electric Current, Resistance, and Ohm's Law

## Resistance and Resistivity

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the concept of resistivity.
2. Use resistivity to calculate the resistance of specified configurations of material.
3. Use the thermal coefficient of resistivity to calculate the change of resistance with temperature.

### Material and Shape Dependence of Resistance

The resistance of an object depends on its shape and the material of which it is composed. The cylindrical resistor in  is easy to analyze, and, by so doing, we can gain insight into the resistance of more complicated shapes. As you might expect, the cylinder’s electric resistance  is directly proportional to its length , similar to the resistance of a pipe to fluid flow. The longer the cylinder, the more collisions charges will make with its atoms. The greater the diameter of the cylinder, the more current it can carry (again similar to the flow of fluid through a pipe). In fact,  is inversely proportional to the cylinder’s cross-sectional area .
For a given shape, the resistance depends on the material of which the object is composed. Different materials offer different resistance to the flow of charge. We define the resistivity  of a substance so that the resistance  of an object is directly proportional to . Resistivity  is an intrinsic property of a material, independent of its shape or size. The resistance  of a uniform cylinder of length , of cross-sectional area , and made of a material with resistivity , is
 gives representative values of . The materials listed in the table are separated into categories of conductors, semiconductors, and insulators, based on broad groupings of resistivities. Conductors have the smallest resistivities, and insulators have the largest; semiconductors have intermediate resistivities. Conductors have varying but large free charge densities, whereas most charges in insulators are bound to atoms and are not free to move. Semiconductors are intermediate, having far fewer free charges than conductors, but having properties that make the number of free charges depend strongly on the type and amount of impurities in the semiconductor. These unique properties of semiconductors are put to use in modern electronics, as will be explored in later chapters.

### Temperature Variation of Resistance

The resistivity of all materials depends on temperature. Some even become superconductors (zero resistivity) at very low temperatures. (See .) Conversely, the resistivity of conductors increases with increasing temperature. Since the atoms vibrate more rapidly and over larger distances at higher temperatures, the electrons moving through a metal make more collisions, effectively making the resistivity higher. Over relatively small temperature changes (about  or less), resistivity  varies with temperature change  as expressed in the following equation
where  is the original resistivity and  is the temperature coefficient of resistivity. (See the values of  in  below.) For larger temperature changes,  may vary or a nonlinear equation may be needed to find . Note that  is positive for metals, meaning their resistivity increases with temperature. Some alloys have been developed specifically to have a small temperature dependence. Manganin (which is made of copper, manganese and nickel), for example, has  close to zero (to three digits on the scale in ), and so its resistivity varies only slightly with temperature. This is useful for making a temperature-independent resistance standard, for example.
Note also that  is negative for the semiconductors listed in , meaning that their resistivity decreases with increasing temperature. They become better conductors at higher temperature, because increased thermal agitation increases the number of free charges available to carry current. This property of decreasing  with temperature is also related to the type and amount of impurities present in the semiconductors.
The resistance of an object also depends on temperature, since  is directly proportional to . For a cylinder we know , and so, if  and  do not change greatly with temperature,  will have the same temperature dependence as . (Examination of the coefficients of linear expansion shows them to be about two orders of magnitude less than typical temperature coefficients of resistivity, and so the effect of temperature on  and  is about two orders of magnitude less than on .) Thus,
is the temperature dependence of the resistance of an object, where  is the original resistance and  is the resistance after a temperature change . Numerous thermometers are based on the effect of temperature on resistance. (See .) One of the most common is the thermistor, a semiconductor crystal with a strong temperature dependence, the resistance of which is measured to obtain its temperature. The device is small, so that it quickly comes into thermal equilibrium with the part of a person it touches.

### Test Prep for AP Courses

### Section Summary

1. The resistance  of a cylinder of length  and cross-sectional area  is , where  is the resistivity of the material.
2. Values of  in  show that materials fall into three groups—conductors, semiconductors, and insulators.
3. Temperature affects resistivity; for relatively small temperature changes , resistivity is , where  is the original resistivity and

    is the temperature coefficient of resistivity.
4.  gives values for , the temperature coefficient of resistivity.
5. The resistance  of an object also varies with temperature: , where  is the original resistance, and

    is the resistance after the temperature change.

### Conceptual Questions

### Problems & Exercises


# Electric Current, Resistance, and Ohm's Law

## Electric Power and Energy

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the power dissipated by a resistor and power supplied by a power supply.
2. Calculate the cost of electricity under various circumstances.

### Power in Electric Circuits

Power is associated by many people with electricity. Knowing that power is the rate of energy use or energy conversion, what is the expression for electric power? Power transmission lines might come to mind. We also think of lightbulbs in terms of their power ratings in watts. Let us compare a 25-W bulb with a 60-W bulb. (See (a).) Since both operate on the same voltage, the 60-W bulb must draw more current to have a greater power rating. Thus the 60-W bulb’s resistance must be lower than that of a 25-W bulb. If we increase voltage, we also increase power. For example, when a 25-W bulb that is designed to operate on 120 V is connected to 240 V, it briefly glows very brightly and then burns out. Precisely how are voltage, current, and resistance related to electric power?
Electric energy depends on both the voltage involved and the charge moved. This is expressed most simply as , where  is the charge moved and  is the voltage (or more precisely, the potential difference the charge moves through). Power is the rate at which energy is moved, and so electric power is
Recognizing that current is  (note that  here), the expression for power becomes
Electric power ( ) is simply the product of current times voltage. Power has familiar units of watts. Since the SI unit for potential energy (PE) is the joule, power has units of joules per second, or watts. Thus, . For example, cars often have one or more auxiliary power outlets with which you can charge a cell phone or other electronic devices. These outlets may be rated at 20 A, so that the circuit can deliver a maximum power
. In some applications, electric power may be expressed as volt-amperes or even kilovolt-amperes (
).
To see the relationship of power to resistance, we combine Ohm’s law with . Substituting  gives . Similarly, substituting  gives . Three expressions for electric power are listed together here for convenience:
Note that the first equation is always valid, whereas the other two can be used only for resistors. In a simple circuit, with one voltage source and a single resistor, the power supplied by the voltage source and that dissipated by the resistor are identical. (In more complicated circuits,  can be the power dissipated by a single device and not the total power in the circuit.)
Different insights can be gained from the three different expressions for electric power. For example,  implies that the lower the resistance connected to a given voltage source, the greater the power delivered. Furthermore, since voltage is squared in , the effect of applying a higher voltage is perhaps greater than expected. Thus, when the voltage is doubled to a 25-W bulb, its power nearly quadruples to about 100 W, burning it out. If the bulb’s resistance remained constant, its power would be exactly 100 W, but at the higher temperature its resistance is higher, too.

### The Cost of Electricity

The more electric appliances you use and the longer they are left on, the higher your electric bill. This familiar fact is based on the relationship between energy and power. You pay for the energy used. Since , we see that
is the energy used by a device using power  for a time interval . For example, the more lightbulbs burning, the greater  used; the longer they are on, the greater  is. The energy unit on electric bills is the kilowatt-hour (), consistent with the relationship . It is easy to estimate the cost of operating electric appliances if you have some idea of their power consumption rate in watts or kilowatts, the time they are on in hours, and the cost per kilowatt-hour for your electric utility. Kilowatt-hours, like all other specialized energy units such as food calories, can be converted to joules. You can prove to yourself that .
The electrical energy ( ) used can be reduced either by reducing the time of use or by reducing the power consumption of that appliance or fixture. This will not only reduce the cost, but it will also result in a reduced impact on the environment. Improvements to lighting are some of the fastest ways to reduce the electrical energy used in a home or business. About 20% of a home’s use of energy goes to lighting, while the number for commercial establishments is closer to 40%. Fluorescent lights are about four times more efficient than incandescent lights—this is true for both the long tubes and the compact fluorescent lights (CFL). (See (b).) Thus, a 60-W incandescent bulb can be replaced by a 15-W CFL, which has the same brightness and color. CFLs have a bent tube inside a globe or a spiral-shaped tube, all connected to a standard screw-in base that fits standard incandescent light sockets. (Original problems with color, flicker, shape, and high initial investment for CFLs have been addressed in recent years.) The heat transfer from these CFLs is less, and they last up to 10 times longer. The significance of an investment in such bulbs is addressed in the next example. New white LED lights (which are clusters of small LED bulbs) are even more efficient (twice that of CFLs) and last 5 times longer than CFLs. However, their cost is still high.

### Test Prep for AP Courses

### Section Summary

1. Electric power  is the rate (in watts) that energy is supplied by a source or dissipated by a device.
2. Three expressions for electrical power are

    and

3. The energy used by a device with a power  over a time  is .

### Conceptual Questions

### Problem Exercises


# Electric Current, Resistance, and Ohm's Law

## Alternating Current versus Direct Current

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the differences and similarities between AC and DC current.
2. Calculate rms voltage, current, and average power.
3. Explain why AC current is used for power transmission.

### Alternating Current

Most of the examples dealt with so far, and particularly those utilizing batteries, have constant voltage sources. Once the current is established, it is thus also a constant. Direct current (DC) is the flow of electric charge in only one direction. It is the steady state of a constant-voltage circuit. Most well-known applications, however, use a time-varying voltage source. Alternating current (AC) is the flow of electric charge that periodically reverses direction. If the source varies periodically, particularly sinusoidally, the circuit is known as an alternating current circuit. Examples include the commercial and residential power that serves so many of our needs.  shows graphs of voltage and current versus time for typical DC and AC power. The AC voltages and frequencies commonly used in homes and businesses vary around the world.
 shows a schematic of a simple circuit with an AC voltage source. The voltage between the terminals fluctuates as shown, with the AC voltage given by
where  is the voltage at time ,  is the peak voltage, and  is the frequency in hertz. For this simple resistance circuit, , and so the AC current is
where  is the current at time , and  is the peak current. For this example, the voltage and current are said to be in phase, as seen in (b).
Current in the resistor alternates back and forth just like the driving voltage, since . If the resistor is a fluorescent light bulb, for example, it brightens and dims 120 times per second as the current repeatedly goes through zero. A 120-Hz flicker is too rapid for your eyes to detect, but if you wave your hand back and forth between your face and a fluorescent light, you will see a stroboscopic effect evidencing AC. The fact that the light output fluctuates means that the power is fluctuating. The power supplied is . Using the expressions for  and  above, we see that the time dependence of power is , as shown in .
We are most often concerned with average power rather than its fluctuations—that 60-W light bulb in your desk lamp has an average power consumption of 60 W, for example. As illustrated in , the average power  is
This is evident from the graph, since the areas above and below the  line are equal, but it can also be proven using trigonometric identities. Similarly, we define an average or rms current  and average or rms voltage  to be, respectively,
and
where rms stands for root mean square, a particular kind of average. In general, to obtain a root mean square, the particular quantity is squared, its mean (or average) is found, and the square root is taken. This is useful for AC, since the average value is zero. Now,
which gives
as stated above. It is standard practice to quote , , and  rather than the peak values. For example, most household electricity is 120 V AC, which means that  is 120 V. The common 10-A circuit breaker will interrupt a sustained  greater than 10 A. Your 1.0-kW microwave oven consumes , and so on. You can think of these rms and average values as the equivalent DC values for a simple resistive circuit.
To summarize, when dealing with AC, Ohm’s law and the equations for power are completely analogous to those for DC, but rms and average values are used for AC. Thus, for AC, Ohm’s law is written
The various expressions for AC power  are
and

### Why Use AC for Power Distribution?

Most large power-distribution systems are AC. Moreover, the power is transmitted at much higher voltages than the 120-V AC (240 V in most parts of the world) we use in homes and on the job. Economies of scale make it cheaper to build a few very large electric power-generation plants than to build numerous small ones. This necessitates sending power long distances, and it is obviously important that energy losses en route be minimized. High voltages can be transmitted with much smaller power losses than low voltages, as we shall see. (See .) For safety reasons, the voltage at the user is reduced to familiar values. The crucial factor is that it is much easier to increase and decrease AC voltages than DC, so AC is used in most large power distribution systems.
It is widely recognized that high voltages pose greater hazards than low voltages. But, in fact, some high voltages, such as those associated with common static electricity, can be harmless. So it is not voltage alone that determines a hazard. It is not so widely recognized that AC shocks are often more harmful than similar DC shocks. Thomas Edison thought that AC shocks were more harmful and set up a DC power-distribution system in New York City in the late 1800s. There were bitter fights, in particular between Edison and George Westinghouse and Nikola Tesla, who were advocating the use of AC in early power-distribution systems. AC has prevailed largely due to transformers and lower power losses with high-voltage transmission.

### Section Summary

1. Direct current  (DC) is the flow of electric current in only one direction. It refers to systems where the source voltage is constant.
2. The voltage source of an alternating current (AC) system puts out , where  is the voltage at time ,  is the peak voltage, and  is the frequency in hertz.
3. In a simple circuit,  and AC current is , where  is the current at time , and  is the peak current.
4. The average AC power  is  .
5. Average (rms) current  and average (rms) voltage  are  and , where rms stands for root mean square.
6. Thus, .
7. Ohm’s law for AC is .
8. Expressions for the average power of an AC circuit  are ,

, and

, analogous to the expressions for DC circuits.

### Conceptual Questions

### Problem Exercises


# Electric Current, Resistance, and Ohm's Law

## Electric Hazards and the Human Body

### Learning Objectives

By the end of this section, you will be able to:
1. Define thermal hazard, shock hazard, and short circuit.
2. Explain what effects various levels of current have on the human body.
There are two known hazards of electricity—thermal and shock. A thermal hazard is one where excessive electric power causes undesired thermal effects, such as starting a fire in the wall of a house. A shock hazard occurs when electric current passes through a person. Shocks range in severity from painful, but otherwise harmless, to heart-stopping lethality. This section considers these hazards and the various factors affecting them in a quantitative manner. Electrical Safety: Systems and Devices will consider systems and devices for preventing electrical hazards.

### Thermal Hazards

Electric power causes undesired heating effects whenever electric energy is converted to thermal energy at a rate faster than it can be safely dissipated. A classic example of this is the short circuit, a low-resistance path between terminals of a voltage source. An example of a short circuit is shown in . Insulation on wires leading to an appliance has worn through, allowing the two wires to come into contact. Such an undesired contact with a high voltage is called a short. Since the resistance of the short, , is very small, the power dissipated in the short, , is very large. For example, if  is 120 V and  is , then the power is 144 kW, much greater than that used by a typical household appliance. Thermal energy delivered at this rate will very quickly raise the temperature of surrounding materials, melting or perhaps igniting them.
One particularly insidious aspect of a short circuit is that its resistance may actually be decreased due to the increase in temperature. This can happen if the short creates ionization. These charged atoms and molecules are free to move and, thus, lower the resistance . Since , the power dissipated in the short rises, possibly causing more ionization, more power, and so on. High voltages, such as the 480-V AC used in some industrial applications, lend themselves to this hazard, because higher voltages create higher initial power production in a short.
Another serious, but less dramatic, thermal hazard occurs when wires supplying power to a user are overloaded with too great a current. As discussed in the previous section, the power dissipated in the supply wires is , where  is the resistance of the wires and  the current flowing through them. If either  or  is too large, the wires overheat. For example, a worn appliance cord (with some of its braided wires broken) may have  rather than the  it should be. If 10.0 A of current passes through the cord, then  is dissipated in the cord—much more than is safe. Similarly, if a wire with a  resistance is meant to carry a few amps, but is instead carrying 100 A, it will severely overheat. The power dissipated in the wire will in that case be . Fuses and circuit breakers are used to limit excessive currents. (See  and .) Each device opens the circuit automatically when a sustained current exceeds safe limits.
Fuses and circuit breakers for typical household voltages and currents are relatively simple to produce, but those for large voltages and currents experience special problems. For example, when a circuit breaker tries to interrupt the flow of high-voltage electricity, a spark can jump across its points that ionizes the air in the gap and allows the current to continue flowing. Large circuit breakers found in power-distribution systems employ insulating gas and even use jets of gas to blow out such sparks. Here AC is safer than DC, since AC current goes through zero 120 times per second, giving a quick opportunity to extinguish these arcs.

### Shock Hazards

Electrical currents through people produce tremendously varied effects. An electrical current can be used to block back pain. The possibility of using electrical current to stimulate muscle action in paralyzed limbs, perhaps allowing paraplegics to walk, is under study. TV dramatizations in which electrical shocks are used to bring a heart attack victim out of ventricular fibrillation (a massively irregular, often fatal, beating of the heart) are more than common. Yet most electrical shock fatalities occur because a current put the heart into fibrillation. A pacemaker uses electrical shocks to stimulate the heart to beat properly. Some fatal shocks do not produce burns, but warts can be safely burned off with electric current (though freezing using liquid nitrogen is now more common). Of course, there are consistent explanations for these disparate effects. The major factors upon which the effects of electrical shock depend are
1. The amount of current
2. The path taken by the current
3. The duration of the shock
4. The frequency  of the current ( for DC)
 gives the effects of electrical shocks as a function of current for a typical accidental shock. The effects are for a shock that passes through the trunk of the body, has a duration of 1 s, and is caused by 60-Hz power.
Our bodies are relatively good conductors due to the water in our bodies. Given that larger currents will flow through sections with lower resistance (to be further discussed in the next chapter), electric currents preferentially flow through paths in the human body that have a minimum resistance in a direct path to earth. The earth is a natural electron sink. Wearing insulating shoes, a requirement in many professions, prohibits a pathway for electrons by providing a large resistance in that path. Whenever working with high-power tools (drills), or in risky situations, ensure that you do not provide a pathway for current flow (especially through the heart).
Very small currents pass harmlessly and unfelt through the body. This happens to you regularly without your knowledge. The threshold of sensation is only 1 mA and, although unpleasant, shocks are apparently harmless for currents less than 5 mA. A great number of safety rules take the 5-mA value for the maximum allowed shock. At 10 to 20 mA and above, the current can stimulate sustained muscular contractions much as regular nerve impulses do. People sometimes say they were knocked across the room by a shock, but what really happened was that certain muscles contracted, propelling them in a manner not of their own choosing. (See (a).) More frightening, and potentially more dangerous, is the “can’t let go” effect illustrated in (b). The muscles that close the fingers are stronger than those that open them, so the hand closes involuntarily on the wire shocking it. This can prolong the shock indefinitely. It can also be a danger to a person trying to rescue the victim, because the rescuer’s hand may close about the victim’s wrist. Usually the best way to help the victim is to give the fist a hard knock/blow/jar with an insulator or to throw an insulator at the fist. Modern electric fences, used in animal enclosures, are now pulsed on and off to allow people who touch them to get free, rendering them less lethal than in the past.
Greater currents may affect the heart. Its electrical patterns can be disrupted, so that it beats irregularly and ineffectively in a condition called “ventricular fibrillation.” This condition often lingers after the shock and is fatal due to a lack of blood circulation. The threshold for ventricular fibrillation is between 100 and 300 mA. At about 300 mA and above, the shock can cause burns, depending on the concentration of current—the more concentrated, the greater the likelihood of burns.
Very large currents cause the heart and diaphragm to contract for the duration of the shock. Both the heart and breathing stop. Interestingly, both often return to normal following the shock. The electrical patterns on the heart are completely erased in a manner that the heart can start afresh with normal beating, as opposed to the permanent disruption caused by smaller currents that can put the heart into ventricular fibrillation. The latter is something like scribbling on a blackboard, whereas the former completely erases it. TV dramatizations of electric shock used to bring a heart attack victim out of ventricular fibrillation also show large paddles. These are used to spread out current passed through the victim to reduce the likelihood of burns.
Current is the major factor determining shock severity (given that other conditions such as path, duration, and frequency are fixed, such as in the table and preceding discussion). A larger voltage is more hazardous, but since , the severity of the shock depends on the combination of voltage and resistance. For example, a person with dry skin has a resistance of about . If he comes into contact with 120-V AC, a current  passes harmlessly through him. The same person soaking wet may have a resistance of  and the same 120 V will produce a current of 12 mA—above the “can’t let go” threshold and potentially dangerous.
Most of the body’s resistance is in its dry skin. When wet, salts go into ion form, lowering the resistance significantly. The interior of the body has a much lower resistance than dry skin because of all the ionic solutions and fluids it contains. If skin resistance is bypassed, such as by an intravenous infusion, a catheter, or exposed pacemaker leads, a person is rendered microshock sensitive. In this condition, currents about 1/1000 those listed in  produce similar effects. During open-heart surgery, currents as small as  can be used to still the heart. Stringent electrical safety requirements in hospitals, particularly in surgery and intensive care, are related to the doubly disadvantaged microshock-sensitive patient. The break in the skin has reduced his resistance, and so the same voltage causes a greater current, and a much smaller current has a greater effect.
Factors other than current that affect the severity of a shock are its path, duration, and AC frequency. Path has obvious consequences. For example, the heart is unaffected by an electric shock through the brain, such as may be used to treat manic depression. And it is a general truth that the longer the duration of a shock, the greater its effects.  presents a graph that illustrates the effects of frequency on a shock. The curves show the minimum current for two different effects, as a function of frequency. The lower the current needed, the more sensitive the body is at that frequency. Ironically, the body is most sensitive to frequencies near the 50- or 60-Hz frequencies in common use. The body is slightly less sensitive for DC (), mildly confirming Edison’s claims that AC presents a greater hazard. At higher and higher frequencies, the body becomes progressively less sensitive to any effects that involve nerves. This is related to the maximum rates at which nerves can fire or be stimulated. At very high frequencies, electrical current travels only on the surface of a person. Thus a wart can be burned off with very high frequency current without causing the heart to stop. (Do not try this at home with 60-Hz AC!) Some of the spectacular demonstrations of electricity, in which high-voltage arcs are passed through the air and over people’s bodies, employ high frequencies and low currents. (See .) Electrical safety devices and techniques are discussed in detail in Electrical Safety: Systems and Devices.

### Section Summary

1. The two types of electric hazards are thermal (excessive power) and shock (current through a person).
2. Shock severity is determined by current, path, duration, and AC frequency.
3.  lists shock hazards as a function of current.
4.  graphs the threshold current for two hazards as a function of frequency.

### Conceptual Questions

### Problem Exercises


# Electric Current, Resistance, and Ohm's Law

## Nerve Conduction–Electrocardiograms

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the process by which electric signals are transmitted along a neuron.
2. Explain the effects myelin sheaths have on signal propagation.
3. Explain what the features of an ECG signal indicate.

### Nerve Conduction

Electric currents in the vastly complex system of billions of nerves in our body allow us to sense the world, control parts of our body, and think. These are representative of the three major functions of nerves. First, nerves carry messages from our sensory organs and others to the central nervous system, consisting of the brain and spinal cord. Second, nerves carry messages from the central nervous system to muscles and other organs. Third, nerves transmit and process signals within the central nervous system. The sheer number of nerve cells and the incredibly greater number of connections between them makes this system the subtle wonder that it is. Nerve conduction is a general term for electrical signals carried by nerve cells. It is one aspect of bioelectricity, or electrical effects in and created by biological systems.
Nerve cells, properly called neurons, look different from other cells—they have tendrils, some of them many centimeters long, connecting them with other cells. (See .) Signals arrive at the cell body across synapses or through dendrites, stimulating the neuron to generate its own signal, sent along its long axon to other nerve or muscle cells. Signals may arrive from many other locations and be transmitted to yet others, conditioning the synapses by use, giving the system its complexity and its ability to learn.
The method by which these electric currents are generated and transmitted is more complex than the simple movement of free charges in a conductor, but it can be understood with principles already discussed in this text. The most important of these are the Coulomb force and diffusion.
 illustrates how a voltage (potential difference) is created across the cell membrane of a neuron in its resting state. This thin membrane separates electrically neutral fluids having differing concentrations of ions, the most important varieties being , , and  (these are sodium, potassium, and chlorine ions with single plus or minus charges as indicated). As discussed in Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes, free ions will diffuse from a region of high concentration to one of low concentration. But the cell membrane is semipermeable, meaning that some ions may cross it while others cannot. In its resting state, the cell membrane is permeable to  and , and impermeable to . Diffusion of  and  thus creates the layers of positive and negative charge on the outside and inside of the membrane. The Coulomb force prevents the ions from diffusing across in their entirety. Once the charge layer has built up, the repulsion of like charges prevents more from moving across, and the attraction of unlike charges prevents more from leaving either side. The result is two layers of charge right on the membrane, with diffusion being balanced by the Coulomb force. A tiny fraction of the charges move across and the fluids remain neutral (other ions are present), while a separation of charge and a voltage have been created across the membrane.
The separation of charge creates a potential difference of 70 to 90 mV across the cell membrane. While this is a small voltage, the resulting electric field () across the only 8-nm-thick membrane is immense (on the order of 11 MV/m!) and has fundamental effects on its structure and permeability. Now, if the exterior of a neuron is taken to be at 0 V, then the interior has a resting potential of about –90 mV. Such voltages are created across the membranes of almost all types of animal cells but are largest in nerve and muscle cells. In fact, fully 25% of the energy used by cells goes toward creating and maintaining these potentials.
Electric currents along the cell membrane are created by any stimulus that changes the membrane’s permeability. The membrane thus temporarily becomes permeable to , which then rushes in, driven both by diffusion and the Coulomb force. This inrush of  first neutralizes the inside membrane, or depolarizes it, and then makes it slightly positive. The depolarization causes the membrane to again become impermeable to , and the movement of  quickly returns the cell to its resting potential, or repolarizes it. This sequence of events results in a voltage pulse, called the action potential. (See .) Only small fractions of the ions move, so that the cell can fire many hundreds of times without depleting the excess concentrations of  and . Eventually, the cell must replenish these ions to maintain the concentration differences that create bioelectricity. This sodium-potassium pump is an example of active transport, wherein cell energy is used to move ions across membranes against diffusion gradients and the Coulomb force.
The action potential is a voltage pulse at one location on a cell membrane. How does it get transmitted along the cell membrane, and in particular down an axon, as a nerve impulse? The answer is that the changing voltage and electric fields affect the permeability of the adjacent cell membrane, so that the same process takes place there. The adjacent membrane depolarizes, affecting the membrane further down, and so on, as illustrated in . Thus the action potential stimulated at one location triggers a nerve impulse that moves slowly (about 1 m/s) along the cell membrane.
Some axons, like that in , are sheathed with myelin, consisting of fat-containing cells.  shows an enlarged view of an axon having myelin sheaths characteristically separated by unmyelinated gaps (called nodes of Ranvier). This arrangement gives the axon a number of interesting properties. Since myelin is an insulator, it prevents signals from jumping between adjacent nerves (cross talk). Additionally, the myelinated regions transmit electrical signals at a very high speed, as an ordinary conductor or resistor would. There is no action potential in the myelinated regions, so that no cell energy is used in them. There is an  signal loss in the myelin, but the signal is regenerated in the gaps, where the voltage pulse triggers the action potential at full voltage. So a myelinated axon transmits a nerve impulse faster, with less energy consumption, and is better protected from cross talk than an unmyelinated one. Not all axons are myelinated, so that cross talk and slow signal transmission are a characteristic of the normal operation of these axons, another variable in the nervous system.
The degeneration or destruction of the myelin sheaths that surround the nerve fibers impairs signal transmission and can lead to numerous neurological effects. One of the most prominent of these diseases comes from the body’s own immune system attacking the myelin in the central nervous system—multiple sclerosis. MS symptoms include fatigue, vision problems, weakness of arms and legs, loss of balance, and tingling or numbness in one’s extremities (neuropathy). It is more apt to strike younger adults, especially females. Causes might come from infection, environmental or geographic affects, or genetics. At the moment there is no known cure for MS.
Most animal cells can fire or create their own action potential. Muscle cells contract when they fire and are often induced to do so by a nerve impulse. In fact, nerve and muscle cells are physiologically similar, and there are even hybrid cells, such as in the heart, that have characteristics of both nerves and muscles. Some animals, like the infamous electric eel (see ), use muscles ganged so that their voltages add in order to create a shock great enough to stun prey.

### Electrocardiograms

Just as nerve impulses are transmitted by depolarization and repolarization of adjacent membrane, the depolarization that causes muscle contraction can also stimulate adjacent muscle cells to depolarize (fire) and contract. Thus, a depolarization wave can be sent across the heart, coordinating its rhythmic contractions and enabling it to perform its vital function of propelling blood through the circulatory system.  is a simplified graphic of a depolarization wave spreading across the heart from the sinoarterial (SA) node, the heart’s natural pacemaker.
An electrocardiogram (ECG) is a record of the voltages created by the wave of depolarization and subsequent repolarization in the heart. (They are also abbreviated EKG.) Voltages between pairs of electrodes placed on the chest are vector components of the voltage wave on the heart. Standard ECGs have 12 or more electrodes, but only three are shown in  for clarity. Decades ago, three-electrode ECGs were performed by placing electrodes on the left and right arms and the left leg. The voltage between the right arm and the left leg is called the lead II potential and is the most often graphed. We shall examine the lead II potential as an indicator of heart-muscle function and see that it is coordinated with arterial blood pressure as well.
Heart function and its four-chamber action are explored in Viscosity and Laminar Flow; Poiseuille’s Law. Basically, the right and left atria receive blood from the body and lungs, respectively, and pump the blood into the ventricles. The right and left ventricles, in turn, pump blood through the lungs and the rest of the body, respectively. Depolarization of the heart muscle causes it to contract. After contraction it is repolarized to ready it for the next beat. The ECG measures components of depolarization and repolarization of the heart muscle and can yield significant information on the functioning and malfunctioning of the heart.
 shows an ECG of the lead II potential and a graph of the corresponding arterial blood pressure. The major features are labeled P, Q, R, S, and T. The P wave is generated by the depolarization and contraction of the atria as they pump blood into the ventricles. The QRS complex is created by the depolarization of the ventricles as they pump blood to the lungs and body. Since the shape of the heart and the path of the depolarization wave are not simple, the QRS complex has this typical shape and time span. The lead II QRS signal also masks the repolarization of the atria, which occur at the same time. Finally, the T wave is generated by the repolarization of the ventricles and is followed by the next P wave in the next heartbeat. Arterial blood pressure varies with each part of the heartbeat, with systolic (maximum) pressure occurring closely after the QRS complex, which signals contraction of the ventricles.
Taken together, the 12 leads of a state-of-the-art ECG can yield a wealth of information about the heart. For example, regions of damaged heart tissue, called infarcts, reflect electrical waves and are apparent in one or more lead potentials. Subtle changes due to slight or gradual damage to the heart are most readily detected by comparing a recent ECG to an older one. This is particularly the case since individual heart shape, size, and orientation can cause variations in ECGs from one individual to another. ECG technology has advanced to the point where a portable ECG monitor can be incorporated into wearable devices and other small objects. See .

### Section Summary

1. Electric potentials in neurons and other cells are created by ionic concentration differences across semipermeable membranes.
2. Stimuli change the permeability and create action potentials that propagate along neurons.
3. Myelin sheaths speed this process and reduce the needed energy input.
4. This process in the heart can be measured with an electrocardiogram (ECG).

### Conceptual Questions

### Problems & Exercises


# Circuits and DC Instruments

## Connection for AP® Courses

Electric circuits are commonplace in our everyday lives. Some circuits are simple, such as those in flashlights while others are extremely complex, such as those used in supercomputers. This chapter takes the topic of electric circuits a step beyond simple circuits by addressing both changes that result from interactions between systems (Big Idea 4) and constraints on such changes due to laws of conservation (Big Idea 5). When the circuit is purely resistive, everything in this chapter applies to both DC and AC. However, matters become more complex when capacitance is involved. We do consider what happens when capacitors are connected to DC voltage sources, but the interaction of capacitors (and other nonresistive devices) with AC sources is left for a later chapter. In addition, a number of important DC instruments, such as meters that measure voltage and current, are covered in this chapter.
Information and examples presented in the chapter examine cause-effect relationships inherent in interactions involving electrical systems. The electrical properties of an electric circuit can change due to other systems (Enduring Understanding 4.E). More specifically, values of currents and potential differences in electric circuits depend on arrangements of individual circuit components (Essential Knowledge 4.E.5). In this chapter several series and parallel combinations of resistors are discussed and their effects on currents and potential differences are analyzed.
In electric circuits the total energy (Enduring Understanding 5.B) and the total electric charge (Enduring Understanding 5.C) are conserved. Kirchoff’s rules describe both, energy conservation (Essential Knowledge 5.B.9) and charge conservation (Essential Knowledge 5.C.3). Energy conservation is discussed in terms of the loop rule which specifies that the potential around any closed circuit path must be zero. Charge conservation is applied as conservation of current by equating the sum of all currents entering a junction to the sum of all currents leaving the junction (also known as the junction rule). Kirchoff’s rules are used to calculate currents and potential differences in circuits that combine resistors in series and parallel, and resistors and capacitors.
The concepts in this chapter support:
Big Idea 4 Interactions between systems can result in changes in those systems.
Enduring Understanding 4.E The electric and magnetic properties of a system can change in response to the presence of, or changes in, other objects or systems.
Essential Knowledge 4.E.5 The values of currents and electric potential differences in an electric circuit are determined by the properties and arrangement of the individual circuit elements such as sources of emf, resistors, and capacitors.
Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws.
Enduring Understanding 5.B The energy of a system is conserved.
Essential Knowledge 5.B.9 Kirchhoff’s loop rule describes conservation of energy in electrical circuits.
Enduring Understanding 5.C The electric charge of a system is conserved.
Essential Knowledge 5.C.3 Kirchhoff’s junction rule describes the conservation of electric charge in electrical circuits. Since charge is conserved, current must be conserved at each junction in the circuit. Examples should include circuits that combine resistors in series and parallel.


# Circuits and DC Instruments

## Resistors in Series and Parallel

### Learning Objectives

By the end of this section, you will be able to:
1. Draw a circuit with resistors in parallel and in series.
2. Calculate the voltage drop of a current across a resistor using Ohm’s law.
3. Contrast the way total resistance is calculated for resistors in series and in parallel.
4. Explain why total resistance of a parallel circuit is less than the smallest resistance of any of the resistors in that circuit.
5. Calculate total resistance of a circuit that contains a mixture of resistors connected in series and in parallel.
Most circuits have more than one component, called a resistor that limits the flow of charge in the circuit. A measure of this limit on charge flow is called resistance. The simplest combinations of resistors are the series and parallel connections illustrated in . The total resistance of a combination of resistors depends on both their individual values and how they are connected.

### Resistors in Series

When are resistors in series? Resistors are in series whenever the flow of charge, called the current, must flow through devices sequentially. For example, if current flows through a person holding a screwdriver and into the Earth, then  in (a) could be the resistance of the screwdriver’s shaft,  the resistance of its handle,  the person’s body resistance, and  the resistance of her shoes.
 shows resistors in series connected to a voltage source. It seems reasonable that the total resistance is the sum of the individual resistances, considering that the current has to pass through each resistor in sequence. (This fact would be an advantage to a person wishing to avoid an electrical shock, who could reduce the current by wearing high-resistance rubber-soled shoes. It could be a disadvantage if one of the resistances were a faulty high-resistance cord to an appliance that would reduce the operating current.)
To verify that resistances in series do indeed add, let us consider the loss of electrical power, called a voltage drop, in each resistor in .
According to Ohm’s law, the voltage drop, , across a resistor when a current flows through it is calculated using the equation , where  equals the current in amps (A) and  is the resistance in ohms . Another way to think of this is that  is the voltage necessary to make a current  flow through a resistance .
So the voltage drop across  is , that across  is , and that across  is . The sum of these voltages equals the voltage output of the source; that is,
This equation is based on the conservation of energy and conservation of charge. Electrical potential energy can be described by the equation , where  is the electric charge and  is the voltage. Thus the energy supplied by the source is , while that dissipated by the resistors is
These energies must be equal, because there is no other source and no other destination for energy in the circuit. Thus, . The charge  cancels, yielding , as stated. (Note that the same amount of charge passes through the battery and each resistor in a given amount of time, since there is no capacitance to store charge, there is no place for charge to leak, and charge is conserved.)
Now substituting the values for the individual voltages gives
Note that for the equivalent single series resistance , we have
This implies that the total or equivalent series resistance  of three resistors is .
This logic is valid in general for any number of resistors in series; thus, the total resistance  of a series connection is
as proposed. Since all of the current must pass through each resistor, it experiences the resistance of each, and resistances in series simply add up.

### Resistors in Parallel

 shows resistors in parallel, wired to a voltage source. Resistors are in parallel when each resistor is connected directly to the voltage source by connecting wires having negligible resistance. Each resistor thus has the full voltage of the source applied to it.
Each resistor draws the same current it would if it alone were connected to the voltage source (provided the voltage source is not overloaded). For example, an automobile’s headlights, radio, and so on, are wired in parallel, so that they utilize the full voltage of the source and can operate completely independently. The same is true in your house, or any building. (See (b).)
To find an expression for the equivalent parallel resistance , let us consider the currents that flow and how they are related to resistance. Since each resistor in the circuit has the full voltage, the currents flowing through the individual resistors are , , and . Conservation of charge implies that the total current  produced by the source is the sum of these currents:
Substituting the expressions for the individual currents gives
Note that Ohm’s law for the equivalent single resistance gives
The terms inside the parentheses in the last two equations must be equal. Generalizing to any number of resistors, the total resistance  of a parallel connection is related to the individual resistances by
This relationship results in a total resistance  that is less than the smallest of the individual resistances. (This is seen in the next example.) When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, and so the total resistance is lower.

### Combinations of Series and Parallel

More complex connections of resistors are sometimes just combinations of series and parallel. These are commonly encountered, especially when wire resistance is considered. In that case, wire resistance is in series with other resistances that are in parallel.
Combinations of series and parallel can be reduced to a single equivalent resistance using the technique illustrated in . Various parts are identified as either series or parallel, reduced to their equivalents, and further reduced until a single resistance is left. The process is more time consuming than difficult.
The simplest combination of series and parallel resistance, shown in , is also the most instructive, since it is found in many applications. For example,  could be the resistance of wires from a car battery to its electrical devices, which are in parallel.  and  could be the starter motor and a passenger compartment light. We have previously assumed that wire resistance is negligible, but, when it is not, it has important effects, as the next example indicates.

### Practical Implications

One implication of this last example is that resistance in wires reduces the current and power delivered to a resistor. If wire resistance is relatively large, as in a worn (or a very long) extension cord, then this loss can be significant. If a large current is drawn, the  drop in the wires can also be significant.
For example, when you are rummaging in the refrigerator and the motor comes on, the refrigerator light dims momentarily. Similarly, you can see the passenger compartment light dim when you start the engine of your car (although this may be due to resistance inside the battery itself).
What is happening in these high-current situations is illustrated in . The device represented by  has a very low resistance, and so when it is switched on, a large current flows. This increased current causes a larger  drop in the wires represented by , reducing the voltage across the light bulb (which is ), which then dims noticeably.

### Test Prep for AP Courses

### Section Summary

1. The total resistance of an electrical circuit with resistors wired in a series is the sum of the individual resistances:
2. Each resistor in a series circuit has the same amount of current flowing through it.
3. The voltage drop, or power dissipation, across each individual resistor in a series is different, and their combined total adds up to the power source input.
4. The total resistance of an electrical circuit with resistors wired in parallel is less than the lowest resistance of any of the components and can be determined using the formula:

5. Each resistor in a parallel circuit has the same full voltage of the source applied to it.
6. The current flowing through each resistor in a parallel circuit is different, depending on the resistance.
7. If a more complex connection of resistors is a combination of series and parallel, it can be reduced to a single equivalent resistance by identifying its various parts as series or parallel, reducing each to its equivalent, and continuing until a single resistance is eventually reached.

### Conceptual Questions

### Problem Exercises

Note: Data taken from figures can be assumed to be accurate to three significant digits.


# Circuits and DC Instruments

## Electromotive Force: Terminal Voltage

### Learning Objectives

By the end of this section, you will be able to:
1. Compare and contrast the voltage and the electromagnetic force of an electric power source.
2. Describe what happens to the terminal voltage, current, and power delivered to a load as internal resistance of the voltage source increases (due to aging of batteries, for example).
3. Explain why it is beneficial to use more than one voltage source connected in parallel.
When you forget to turn off your car lights, they slowly dim as the battery runs down. Why don’t they simply blink off when the battery’s energy is gone? Their gradual dimming implies that battery output voltage decreases as the battery is depleted.
Furthermore, if you connect an excessive number of 12-V lights in parallel to a car battery, they will be dim even when the battery is fresh and even if the wires to the lights have very low resistance. This implies that the battery’s output voltage is reduced by the overload.
The reason for the decrease in output voltage for depleted or overloaded batteries is that all voltage sources have two fundamental parts—a source of electrical energy and an internal resistance. Let us examine both.

### Electromotive Force

You can think of many different types of voltage sources. Batteries themselves come in many varieties. There are many types of mechanical/electrical generators, driven by many different energy sources, ranging from nuclear to wind. Solar cells create voltages directly from light, while thermoelectric devices create voltage from temperature differences.
A few voltage sources are shown in . All such devices create a potential difference and can supply current if connected to a resistance. On the small scale, the potential difference creates an electric field that exerts force on charges, causing current. We thus use the name electromotive force, abbreviated emf.
Emf is not a force at all; it is a special type of potential difference. To be precise, the electromotive force (emf) is the potential difference of a source when no current is flowing. Units of emf are volts.
Electromotive force is directly related to the source of potential difference, such as the particular combination of chemicals in a battery. However, emf differs from the voltage output of the device when current flows. The voltage across the terminals of a battery, for example, is less than the emf when the battery supplies current, and it declines further as the battery is depleted or loaded down. However, if the device’s output voltage can be measured without drawing current, then output voltage will equal emf (even for a very depleted battery).

### Internal Resistance

As noted before, a 12-V truck battery is physically larger, contains more charge and energy, and can deliver a larger current than a 12-V motorcycle battery. Both are lead-acid batteries with identical emf, but, because of its size, the truck battery has a smaller internal resistance . Internal resistance is the inherent resistance to the flow of current within the source itself.
 is a schematic representation of the two fundamental parts of any voltage source. The emf (represented by a script E in the figure) and internal resistance  are in series. The smaller the internal resistance for a given emf, the more current and the more power the source can supply.
The internal resistance  can behave in complex ways. As noted,  increases as a battery is depleted. But internal resistance may also depend on the magnitude and direction of the current through a voltage source, its temperature, and even its history. The internal resistance of rechargeable nickel-cadmium cells, for example, depends on how many times and how deeply they have been depleted.
Why are the chemicals able to produce a unique potential difference? Quantum mechanical descriptions of molecules, which take into account the types of atoms and numbers of electrons in them, are able to predict the energy states they can have and the energies of reactions between them.
In the case of a lead-acid battery, an energy of 2 eV is given to each electron sent to the anode. Voltage is defined as the electrical potential energy divided by charge: . An electron volt is the energy given to a single electron by a voltage of 1 V. So the voltage here is 2 V, since 2 eV is given to each electron. It is the energy produced in each molecular reaction that produces the voltage. A different reaction produces a different energy and, hence, a different voltage.

### Terminal Voltage

The voltage output of a device is measured across its terminals and, thus, is called its terminal voltage. Terminal voltage is given by
where  is the internal resistance and  is the current flowing at the time of the measurement.
 is positive if current flows away from the positive terminal, as shown in . You can see that the larger the current, the smaller the terminal voltage. And it is likewise true that the larger the internal resistance, the smaller the terminal voltage.
Suppose a load resistance  is connected to a voltage source, as in . Since the resistances are in series, the total resistance in the circuit is . Thus the current is given by Ohm’s law to be
We see from this expression that the smaller the internal resistance , the greater the current the voltage source supplies to its load . As batteries are depleted,  increases. If  becomes a significant fraction of the load resistance, then the current is significantly reduced, as the following example illustrates.
Battery testers, such as those in , use small load resistors to intentionally draw current to determine whether the terminal voltage drops below an acceptable level. They really test the internal resistance of the battery. If internal resistance is high, the battery is weak, as evidenced by its low terminal voltage.
Some batteries can be recharged by passing a current through them in the direction opposite to the current they supply to a resistance. This is done routinely in cars and batteries for small electrical appliances and electronic devices, and is represented pictorially in . The voltage output of the battery charger must be greater than the emf of the battery to reverse current through it. This will cause the terminal voltage of the battery to be greater than the emf, since , and  is now negative.

### Multiple Voltage Sources

There are two voltage sources when a battery charger is used. Voltage sources connected in series are relatively simple. When voltage sources are in series, their internal resistances add and their emfs add algebraically. (See .) Series connections of voltage sources are common—for example, in flashlights, toys, and other appliances. Usually, the cells are in series in order to produce a larger total emf.
But if the cells oppose one another, such as when one is put into an appliance backward, the total emf is less, since it is the algebraic sum of the individual emfs.
A battery is a multiple connection of voltaic cells, as shown in . The disadvantage of series connections of cells is that their internal resistances add. One of the authors once owned a 1957 MGA that had two 6-V batteries in series, rather than a single 12-V battery. This arrangement produced a large internal resistance that caused him many problems in starting the engine.
If the series connection of two voltage sources is made into a complete circuit with the emfs in opposition, then a current of magnitude  flows. See , for example, which shows a circuit exactly analogous to the battery charger discussed above. If two voltage sources in series with emfs in the same sense are connected to a load , as in , then

 flows.
 shows two voltage sources with identical emfs in parallel and connected to a load resistance. In this simple case, the total emf is the same as the individual emfs. But the total internal resistance is reduced, since the internal resistances are in parallel. The parallel connection thus can produce a larger current.
Here,  flows through the load, and  is less than those of the individual batteries. For example, some diesel-powered cars use two 12-V batteries in parallel; they produce a total emf of 12 V but can deliver the larger current needed to start a diesel engine.

### Animals as Electrical Detectors

A number of animals both produce and detect electrical signals. Fish, sharks, platypuses, and echidnas (spiny anteaters) all detect electric fields generated by nerve activity in prey. Electric eels produce their own emf through biological cells (electric organs) called electroplaques, which are arranged in both series and parallel as a set of batteries.
Electroplaques are flat, disk-like cells; those of the electric eel have a voltage of 0.15 V across each one. These cells are usually located toward the head or tail of the animal, although in the case of the electric eel, they are found along the entire body. The electroplaques in the South American eel are arranged in 140 rows, with each row stretching horizontally along the body and containing 5,000 electroplaques. This can yield an emf of approximately 600 V, and a current of 1 A—deadly.
The mechanism for detection of external electric fields is similar to that for producing nerve signals in the cell through depolarization and repolarization—the movement of ions across the cell membrane. Within the fish, weak electric fields in the water produce a current in a gel-filled canal that runs from the skin to sensing cells, producing a nerve signal. The Australian platypus, one of the very few mammals that lay eggs, can detect fields of 30 , while sharks have been found to be able to sense a field in their snouts as small as 100  (). Electric eels use their own electric fields produced by the electroplaques to stun their prey or enemies.

### Solar Cell Arrays

Another example dealing with multiple voltage sources is that of combinations of solar cells—wired in both series and parallel combinations to yield a desired voltage and current. Photovoltaic generation (PV), the conversion of sunlight directly into electricity, is based upon the photoelectric effect, in which photons hitting the surface of a solar cell create an electric current in the cell.
Most solar cells are made from pure silicon—either as single-crystal silicon, or as a thin film of silicon deposited upon a glass or metal backing. Most single cells have a voltage output of about 0.5 V, while the current output is a function of the amount of sunlight upon the cell (the incident solar radiation—the insolation). Under bright noon sunlight, a current of about  of cell surface area is produced by typical single-crystal cells.
Individual solar cells are connected electrically in modules to meet electrical-energy needs. They can be wired together in series or in parallel—connected like the batteries discussed earlier. A solar-cell array or module usually consists of between 36 and 72 cells, with a power output of 50 W to 140 W.
The output of the solar cells is direct current. For most uses in a home, AC is required, so a device called an inverter must be used to convert the DC to AC. Any extra output can then be passed on to the outside electrical grid for sale to the utility.

### Test Prep for AP Courses

### Section Summary

1. All voltage sources have two fundamental parts—a source of electrical energy that has a characteristic electromotive force (emf), and an internal resistance .
2. The emf is the potential difference of a source when no current is flowing.
3. The numerical value of the emf depends on the source of potential difference.
4. The internal resistance  of a voltage source affects the output voltage when a current flows.
5. The voltage output of a device is called its terminal voltage  and is given by , where  is the electric current and is positive when flowing away from the positive terminal of the voltage source.
6. When multiple voltage sources are in series, their internal resistances add and their emfs add algebraically.
7. Solar cells can be wired in series or parallel to provide increased voltage or current, respectively.

### Conceptual Questions

### Problem Exercises


# Circuits and DC Instruments

## Kirchhoff’s Rules

### Learning Objectives

By the end of this section, you will be able to:
1. Analyze a complex circuit using Kirchhoff’s rules, using the conventions for determining the correct signs of various terms.
Many complex circuits, such as the one in , cannot be analyzed with the series-parallel techniques developed in Resistors in Series and Parallel and Electromotive Force: Terminal Voltage. There are, however, two circuit analysis rules that can be used to analyze any circuit, simple or complex. These rules are special cases of the laws of conservation of charge and conservation of energy. The rules are known as Kirchhoff’s rules, after their inventor Gustav Kirchhoff (1824–1887).
Explanations of the two rules will now be given, followed by problem-solving hints for applying Kirchhoff’s rules, and a worked example that uses them.

### Kirchhoff’s First Rule

Kirchhoff’s first rule (the junction rule) is an application of the conservation of charge to a junction; it is illustrated in . Current is the flow of charge, and charge is conserved; thus, whatever charge flows into the junction must flow out. Kirchhoff’s first rule requires that  (see figure). Equations like this can and will be used to analyze circuits and to solve circuit problems.

### Kirchhoff’s Second Rule

Kirchhoff’s second rule (the loop rule) is an application of conservation of energy. The loop rule is stated in terms of potential, , rather than potential energy, but the two are related since . Recall that emf is the potential difference of a source when no current is flowing.  In a closed loop, whatever energy is supplied by emf must be transferred into other forms by devices in the loop, since there are no other ways in which energy can be transferred into or out of the circuit.  illustrates the changes in potential in a simple series circuit loop.
Kirchhoff’s second rule requires . Rearranged, this is , which means the emf equals the sum of the  (voltage) drops in the loop.

### Applying Kirchhoff’s Rules

By applying Kirchhoff’s rules, we generate equations that allow us to find the unknowns in circuits. The unknowns may be currents, emfs, or resistances. Each time a rule is applied, an equation is produced. If there are as many independent equations as unknowns, then the problem can be solved. There are two decisions you must make when applying Kirchhoff’s rules. These decisions determine the signs of various quantities in the equations you obtain from applying the rules.
1. When applying Kirchhoff’s first rule, the junction rule, you must label the current in each branch and decide in what direction it is going. For example, in , , and , currents are labeled , , , and , and arrows indicate their directions. There is no risk here, for if you choose the wrong direction, the current will be of the correct magnitude but negative.
2. When applying Kirchhoff’s second rule, the loop rule, you must identify a closed loop and decide in which direction to go around it, clockwise or counterclockwise. For example, in  the loop was traversed in the same direction as the current (clockwise). Again, there is no risk; going around the circuit in the opposite direction reverses the sign of every term in the equation, which is like multiplying both sides of the equation by
 and the following points will help you get the plus or minus signs right when applying the loop rule. Note that the resistors and emfs are traversed by going from a to b. In many circuits, it will be necessary to construct more than one loop. In traversing each loop, one needs to be consistent for the sign of the change in potential. (See .)
1. When a resistor is traversed in the same direction as the current, the change in potential is . (See .)
2. When a resistor is traversed in the direction opposite to the current, the change in potential is . (See .)
3. When an emf is traversed from  to + (the same direction it moves positive charge), the change in potential is +emf. (See .)
4. When an emf is traversed from + to  (opposite to the direction it moves positive charge), the change in potential is

          emf. (See .)
The material in this section is correct in theory. We should be able to verify it by making measurements of current and voltage. In fact, some of the devices used to make such measurements are straightforward applications of the principles covered so far and are explored in the next modules. As we shall see, a very basic, even profound, fact results—making a measurement alters the quantity being measured.

### Test Prep for AP Courses

### Section Summary

1. Kirchhoff’s rules can be used to analyze any circuit, simple or complex.
2. Kirchhoff’s first rule—the junction rule: The sum of all currents entering a junction must equal the sum of all currents leaving the junction.
3. Kirchhoff’s second rule—the loop rule: The algebraic sum of changes in potential around any closed circuit path (loop) must be zero.
4. The two rules are based, respectively, on the laws of conservation of charge and energy.
5. When calculating potential and current using Kirchhoff’s rules, a set of conventions must be followed for determining the correct signs of various terms.
6. The simpler series and parallel rules are special cases of Kirchhoff’s rules.

### Conceptual Questions

### Problem Exercises


# Circuits and DC Instruments

## DC Voltmeters and Ammeters

### Learning Objectives

By the end of this section, you will be able to:
1. Explain why a voltmeter must be connected in parallel with the circuit.
2. Draw a diagram showing an ammeter correctly connected in a circuit.
3. Describe how a galvanometer can be used as either a voltmeter or an ammeter.
4. Find the resistance that must be placed in series with a galvanometer to allow it to be used as a voltmeter with a given reading.
5. Explain why measuring the voltage or current in a circuit can never be exact.
Voltmeters measure voltage, whereas ammeters measure current. Some of the meters in automobile dashboards, digital cameras, cell phones, and tuner-amplifiers are voltmeters or ammeters. (See .) The internal construction of the simplest of these meters and how they are connected to the system they monitor give further insight into applications of series and parallel connections.
Voltmeters are connected in parallel with whatever device’s voltage is to be measured. A parallel connection is used because objects in parallel experience the same potential difference. (See , where the voltmeter is represented by the symbol V.)
Ammeters are connected in series with whatever device’s current is to be measured. A series connection is used because objects in series have the same current passing through them. (See , where the ammeter is represented by the symbol A.)

### Analog Meters: Galvanometers

Analog meters have a needle that swivels to point at numbers on a scale, as opposed to digital meters, which have numerical readouts similar to a hand-held calculator. The heart of most analog meters is a device called a galvanometer, denoted by G. Current flow through a galvanometer, , produces a proportional needle deflection. (This deflection is due to the force of a magnetic field upon a current-carrying wire.)
The two crucial characteristics of a given galvanometer are its resistance and current sensitivity. Current sensitivity is the current that gives a full-scale deflection of the galvanometer’s needle, the maximum current that the instrument can measure. For example, a galvanometer with a current sensitivity of  has a maximum deflection of its needle when  flows through it, reads half-scale when  flows through it, and so on.
If such a galvanometer has a  resistance, then a voltage of only  produces a full-scale reading. By connecting resistors to this galvanometer in different ways, you can use it as either a voltmeter or ammeter that can measure a broad range of voltages or currents.

### Galvanometer as Voltmeter

 shows how a galvanometer can be used as a voltmeter by connecting it in series with a large resistance, . The value of the resistance  is determined by the maximum voltage to be measured. Suppose you want 10 V to produce a full-scale deflection of a voltmeter containing a  galvanometer with a  sensitivity. Then 10 V applied to the meter must produce a current of . The total resistance must be
( is so large that the galvanometer resistance, , is nearly negligible.) Note that 5 V applied to this voltmeter produces a half-scale deflection by producing a  current through the meter, and so the voltmeter’s reading is proportional to voltage as desired.
This voltmeter would not be useful for voltages less than about half a volt, because the meter deflection would be small and difficult to read accurately. For other voltage ranges, other resistances are placed in series with the galvanometer. Many meters have a choice of scales. That choice involves switching an appropriate resistance into series with the galvanometer.

### Galvanometer as Ammeter

The same galvanometer can also be made into an ammeter by placing it in parallel with a small resistance , often called the shunt resistance, as shown in . Since the shunt resistance is small, most of the current passes through it, allowing an ammeter to measure currents much greater than those producing a full-scale deflection of the galvanometer.
Suppose, for example, an ammeter is needed that gives a full-scale deflection for 1.0 A, and contains the same  galvanometer with its  sensitivity. Since  and  are in parallel, the voltage across them is the same.
These  drops are  so that . Solving for , and noting that  is  and  is 0.999950 A, we have

### Taking Measurements Alters the Circuit

When you use a voltmeter or ammeter, you are connecting another resistor to an existing circuit and, thus, altering the circuit. Ideally, voltmeters and ammeters do not appreciably affect the circuit, but it is instructive to examine the circumstances under which they do or do not interfere.
First, consider the voltmeter, which is always placed in parallel with the device being measured. Very little current flows through the voltmeter if its resistance is a few orders of magnitude greater than the device, and so the circuit is not appreciably affected. (See (a).) (A large resistance in parallel with a small one has a combined resistance essentially equal to the small one.) If, however, the voltmeter’s resistance is comparable to that of the device being measured, then the two in parallel have a smaller resistance, appreciably affecting the circuit. (See (b).) The voltage across the device is not the same as when the voltmeter is out of the circuit.
An ammeter is placed in series in the branch of the circuit being measured, so that its resistance adds to that branch. Normally, the ammeter’s resistance is very small compared with the resistances of the devices in the circuit, and so the extra resistance is negligible. (See (a).) However, if very small load resistances are involved, or if the ammeter is not as low in resistance as it should be, then the total series resistance is significantly greater, and the current in the branch being measured is reduced. (See (b).)
A practical problem can occur if the ammeter is connected incorrectly. If it was put in parallel with the resistor to measure the current in it, you could possibly damage the meter; the low resistance of the ammeter would allow most of the current in the circuit to go through the galvanometer, and this current would be larger since the effective resistance is smaller.
One solution to the problem of voltmeters and ammeters interfering with the circuits being measured is to use galvanometers with greater sensitivity. This allows construction of voltmeters with greater resistance and ammeters with smaller resistance than when less sensitive galvanometers are used.
There are practical limits to galvanometer sensitivity, but it is possible to get analog meters that make measurements accurate to a few percent. Note that the inaccuracy comes from altering the circuit, not from a fault in the meter.

### Section Summary

1. Voltmeters measure voltage, and ammeters measure current.
2. A voltmeter is placed in parallel with the voltage source to receive full voltage and must have a large resistance to limit its effect on the circuit.
3. An ammeter is placed in series to get the full current flowing through a branch and must have a small resistance to limit its effect on the circuit.
4. Both can be based on the combination of a resistor and a galvanometer, a device that gives an analog reading of current.
5. Standard voltmeters and ammeters alter the circuit being measured and are thus limited in accuracy.

### Conceptual Questions

### Problem Exercises


# Circuits and DC Instruments

## Null Measurements

### Learning Objectives

By the end of this section, you will be able to:
1. Explain why a null measurement device is more accurate than a standard voltmeter or ammeter.
2. Demonstrate how a Wheatstone bridge can be used to accurately calculate the resistance in a circuit.
Standard measurements of voltage and current alter the circuit being measured, introducing uncertainties in the measurements. Voltmeters draw some extra current, whereas ammeters reduce current flow. Null measurements balance voltages so that there is no current flowing through the measuring device and, therefore, no alteration of the circuit being measured.
Null measurements are generally more accurate but are also more complex than the use of standard voltmeters and ammeters, and they still have limits to their precision. In this module, we shall consider a few specific types of null measurements, because they are common and interesting, and they further illuminate principles of electric circuits.

### The Potentiometer

Suppose you wish to measure the emf of a battery. Consider what happens if you connect the battery directly to a standard voltmeter as shown in . (Once we note the problems with this measurement, we will examine a null measurement that improves accuracy.) As discussed before, the actual quantity measured is the terminal voltage , which is related to the emf of the battery by , where  is the current that flows and  is the internal resistance of the battery.
The emf could be accurately calculated if  were very accurately known, but it is usually not. If the current  could be made zero, then , and so emf could be directly measured. However, standard voltmeters need a current to operate; thus, another technique is needed.
A potentiometer is a null measurement device for measuring potentials (voltages). (See .) A voltage source is connected to a resistor  say, a long wire, and passes a constant current through it. There is a steady drop in potential (an  drop) along the wire, so that a variable potential can be obtained by making contact at varying locations along the wire.
(b) shows an unknown  (represented by script   in the figure) connected in series with a galvanometer. Note that  opposes the other voltage source. The location of the contact point (see the arrow on the drawing) is adjusted until the galvanometer reads zero. When the galvanometer reads zero, , where  is the resistance of the section of wire up to the contact point. Since no current flows through the galvanometer, none flows through the unknown emf, and so  is directly sensed.
Now, a very precisely known standard  is substituted for , and the contact point is adjusted until the galvanometer again reads zero, so that . In both cases, no current passes through the galvanometer, and so the current  through the long wire is the same. Upon taking the ratio ,  cancels, giving
Solving for  gives
Because a long uniform wire is used for , the ratio of resistances  is the same as the ratio of the lengths of wire that zero the galvanometer for each emf. The three quantities on the right-hand side of the equation are now known or measured, and  can be calculated. The uncertainty in this calculation can be considerably smaller than when using a voltmeter directly, but it is not zero. There is always some uncertainty in the ratio of resistances  and in the standard . Furthermore, it is not possible to tell when the galvanometer reads exactly zero, which introduces error into both  and , and may also affect the current .

### Resistance Measurements and the Wheatstone Bridge

There is a variety of so-called ohmmeters that purport to measure resistance. What the most common ohmmeters actually do is to apply a voltage to a resistance, measure the current, and calculate the resistance using Ohm’s law. Their readout is this calculated resistance. Two configurations for ohmmeters using standard voltmeters and ammeters are shown in . Such configurations are limited in accuracy, because the meters alter both the voltage applied to the resistor and the current that flows through it.
The Wheatstone bridge is a null measurement device for calculating resistance by balancing potential drops in a circuit. (See .) The device is called a bridge because the galvanometer forms a bridge between two branches. A variety of bridge devices are used to make null measurements in circuits.
Resistors  and  are precisely known, while the arrow through  indicates that it is a variable resistance. The value of  can be precisely read. With the unknown resistance  in the circuit,  is adjusted until the galvanometer reads zero. The potential difference between points b and d is then zero, meaning that b and d are at the same potential. With no current running through the galvanometer, it has no effect on the rest of the circuit. So the branches abc and adc are in parallel, and each branch has the full voltage of the source. That is, the  drops along abc and adc are the same. Since b and d are at the same potential, the  drop along ad must equal the  drop along ab. Thus,
Again, since b and d are at the same potential, the  drop along dc must equal the  drop along bc. Thus,
Taking the ratio of these last two expressions gives
Canceling the currents and solving for Rx yields
This equation is used to calculate the unknown resistance when current through the galvanometer is zero. This method can be very accurate (often to four significant digits), but it is limited by two factors. First, it is not possible to get the current through the galvanometer to be exactly zero. Second, there are always uncertainties in , , and , which contribute to the uncertainty in .

### Section Summary

1. Null measurement techniques achieve greater accuracy by balancing a circuit so that no current flows through the measuring device.
2. One such device, for determining voltage, is a potentiometer.
3. Another null measurement device, for determining resistance, is the Wheatstone bridge.
4. Other physical quantities can also be measured with null measurement techniques.

### Conceptual questions

### Problem Exercises


# Circuits and DC Instruments

## DC Circuits Containing Resistors and Capacitors

When you use a flash camera, it takes a few seconds to charge the capacitor that powers the flash. The light flash discharges the capacitor in a tiny fraction of a second. Why does charging take longer than discharging? This question and a number of other phenomena that involve charging and discharging capacitors are discussed in this module.

### RC Circuits

An  is one containing a resistor R and a capacitor C. The capacitor is an electrical component that stores electric charge.
 shows a simple RC circuit that employs a DC (direct current) voltage source. The capacitor is initially uncharged. As soon as the switch is closed, current flows to and from the initially uncharged capacitor. As charge increases on the capacitor plates, there is increasing opposition to the flow of charge by the repulsion of like charges on each plate.
In terms of voltage, this is because voltage across the capacitor is given by , where  is the amount of charge stored on each plate and  is the capacitance. This voltage opposes the battery, growing from zero to the maximum emf when fully charged. The current thus decreases from its initial value of  to zero as the voltage on the capacitor reaches the same value as the emf. When there is no current, there is no  drop, and so the voltage on the capacitor must then equal the emf of the voltage source. This can also be explained with Kirchhoff’s second rule (the loop rule), discussed in Kirchhoff’s Rules, which says that the algebraic sum of changes in potential around any closed loop must be zero.
The initial current is , because all of the  drop is in the resistance. Therefore, the smaller the resistance, the faster a given capacitor will be charged. Note that the internal resistance of the voltage source is included in , as are the resistances of the capacitor and the connecting wires. In the flash camera scenario above, when the batteries powering the camera begin to wear out, their internal resistance rises, reducing the current and lengthening the time it takes to get ready for the next flash.
Voltage on the capacitor is initially zero and rises rapidly at first, since the initial current is a maximum. (b) shows a graph of capacitor voltage versus time () starting when the switch is closed at . The voltage approaches emf asymptotically, since the closer it gets to emf the less current flows. The equation for voltage versus time when charging a capacitor  through a resistor , derived using calculus, is
where  is the voltage across the capacitor, emf is equal to the emf of the DC voltage source, and the exponential e = 2.718 … is the base of the natural logarithm. Note that the units of  are seconds. We define
where  (the Greek letter tau) is called the time constant for an  circuit. As noted before, a small resistance  allows the capacitor to charge faster. This is reasonable, since a larger current flows through a smaller resistance. It is also reasonable that the smaller the capacitor , the less time needed to charge it. Both factors are contained in .
More quantitatively, consider what happens when . Then the voltage on the capacitor is
This means that in the time , the voltage rises to 0.632 of its final value. The voltage will rise 0.632 of the remainder in the next time . It is a characteristic of the exponential function that the final value is never reached, but 0.632 of the remainder to that value is achieved in every time, . In just a few multiples of the time constant , then, the final value is very nearly achieved, as the graph in (b) illustrates.

### Discharging a Capacitor

Discharging a capacitor through a resistor proceeds in a similar fashion, as  illustrates. Initially, the current is , driven by the initial voltage  on the capacitor. As the voltage decreases, the current and hence the rate of discharge decreases, implying another exponential formula for . Using calculus, the voltage  on a capacitor  being discharged through a resistor  is found to be
The graph in (b) is an example of this exponential decay. Again, the time constant is . A small resistance  allows the capacitor to discharge in a small time, since the current is larger. Similarly, a small capacitance requires less time to discharge, since less charge is stored. In the first time interval  after the switch is closed, the voltage falls to 0.368 of its initial value, since .
During each successive time , the voltage falls to 0.368 of its preceding value. In a few multiples of , the voltage becomes very close to zero, as indicated by the graph in (b).
Now we can explain why the flash camera in our scenario takes so much longer to charge than discharge; the resistance while charging is significantly greater than while discharging. The internal resistance of the battery accounts for most of the resistance while charging. As the battery ages, the increasing internal resistance makes the charging process even slower. (You may have noticed this.)
The flash discharge is through a low-resistance ionized gas in the flash tube and proceeds very rapidly. Flash photographs, such as in , can capture a brief instant of a rapid motion because the flash can be less than a microsecond in duration. Such flashes can be made extremely intense.
During World War II, nighttime reconnaissance photographs were made from the air with a single flash illuminating more than a square kilometer of enemy territory. The brevity of the flash eliminated blurring due to the surveillance aircraft’s motion. Today, an important use of intense flash lamps is to pump energy into a laser. The short intense flash can rapidly energize a laser and allow it to reemit the energy in another form.

### RC Circuits for Timing

 circuits are commonly used for timing purposes. A mundane example of this is found in the ubiquitous intermittent wiper systems of modern cars. The time between wipes is varied by adjusting the resistance in an  circuit. Another example of an  circuit is found in novelty jewelry, Halloween costumes, and various toys that have battery-powered flashing lights. (See  for a timing circuit.)
A more crucial use of  circuits for timing purposes is in the artificial pacemaker, used to control heart rate. The heart rate is normally controlled by electrical signals generated by the sino-atrial (SA) node, which is on the wall of the right atrium chamber. This causes the muscles to contract and pump blood. Sometimes the heart rhythm is abnormal and the heartbeat is too high or too low.
The artificial pacemaker is inserted near the heart to provide electrical signals to the heart when needed with the appropriate time constant. Pacemakers have sensors that detect body motion and breathing to increase the heart rate during exercise to meet the body’s increased needs for blood and oxygen.

### Test Prep for AP Courses

### Section Summary

1. An  circuit is one that has both a resistor and a capacitor.
2. The time constant  for an  circuit is .
3. When an initially uncharged ( at ) capacitor in series with a resistor is charged by a DC voltage source, the voltage rises, asymptotically approaching the emf of the voltage source; as a function of time,

4. Within the span of each time constant , the voltage rises by 0.632 of the remaining value, approaching the final voltage asymptotically.
5. If a capacitor with an initial voltage  is discharged through a resistor starting at , then its voltage decreases exponentially as given by

6. In each time constant , the voltage falls by 0.368 of its  remaining initial value, approaching zero asymptotically.

### Conceptual questions

### Problem Exercises