215. DC Voltmeters and AmmetersLearning Objectives
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 Figure 2, 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 Figure 3, where the ammeter is represented by the symbol A.) Analog Meters: Galvanometers
The two crucial characteristics of a given galvanometer are its resistance and current sensitivity. If such a galvanometer has a $2\text{5}\Omega $ resistance, then a voltage of only $V=\text{IR}=\left(\text{50 \mu A}\right)\left(\text{25 \Omega}\right)=1\text{.}\text{25 mV}$ produces a fullscale 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 VoltmeterFigure 4 shows how a galvanometer can be used as a voltmeter by connecting it in series with a large resistance, $R$. The value of the resistance $R$ is determined by the maximum voltage to be measured. Suppose you want 10 V to produce a fullscale deflection of a voltmeter containing a $2\text{5\Omega}$ galvanometer with a $\text{50\mu A}$ sensitivity. Then 10 V applied to the meter must produce a current of $\text{50 \mu A}$. The total resistance must be $${R}_{\text{tot}}=R+r=\frac{V}{I}=\frac{\text{10}\phantom{\rule{0.25em}{0ex}}\text{V}}{\text{50 \mu A}}=\text{200}\phantom{\rule{0.25em}{0ex}}\text{k}\Omega ,\; or$$ $$R={R}_{\text{tot}}r=\text{200 k\Omega}\text{25}\phantom{\rule{0.25em}{0ex}}\Omega \approx \text{200}\phantom{\rule{0.25em}{0ex}}\text{k}\Omega .$$($R$ is so large that the galvanometer resistance, $r$, is nearly negligible.) Note that 5 V applied to this voltmeter produces a halfscale deflection by producing a $2\text{5\mu A}$ 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 AmmeterThe same galvanometer can also be made into an ammeter by placing it in parallel with a small resistance $R$, often called the Suppose, for example, an ammeter is needed that gives a fullscale deflection for 1.0 A, and contains the same $2\text{5}\Omega $ galvanometer with its $\text{50\mu A}$ sensitivity. Since $R$ and $r$ are in parallel, the voltage across them is the same. These $\text{IR}$ drops are $\text{IR}={I}_{\text{G}}r$ so that $\text{IR}=\frac{{I}_{\text{G}}}{I}=\frac{R}{r}$. Solving for $R$, and noting that ${I}_{\text{G}}$ is $\text{50 \mu A}$ and $I$ is 0.999950 A, we have $$R=r\frac{{I}_{\text{G}}}{I}=(\text{25}\phantom{\rule{0.25em}{0ex}}\Omega )\frac{\text{50 \mu A}}{0\text{.}\text{999950 A}}=1\text{.}\text{25}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\Omega .$$Taking Measurements Alters the CircuitWhen 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 Figure 6(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 Figure 6(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 Figure 7(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 Figure 7(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. Connections: Limits to KnowledgeMaking a measurement alters the system being measured in a manner that produces uncertainty in the measurement. For macroscopic systems, such as the circuits discussed in this module, the alteration can usually be made negligibly small, but it cannot be eliminated entirely. For submicroscopic systems, such as atoms, nuclei, and smaller particles, measurement alters the system in a manner that cannot be made arbitrarily small. This actually limits knowledge of the system—even limiting what nature can know about itself. We shall see profound implications of this when the Heisenberg uncertainty principle is discussed in the modules on quantum mechanics. There is another measurement technique based on drawing no current at all and, hence, not altering the circuit at all. These are called null measurements and are the topic of Null Measurements. Digital meters that employ solidstate electronics and null measurements can attain accuracies of one part in ${\text{10}}^{6}$. Check Your UnderstandingDigital meters are able to detect smaller currents than analog meters employing galvanometers. How does this explain their ability to measure voltage and current more accurately than analog meters? Show/Hide Solution SolutionSince digital meters require less current than analog meters, they alter the circuit less than analog meters. Their resistance as a voltmeter can be far greater than an analog meter, and their resistance as an ammeter can be far less than an analog meter. Consult Figure 2 and Figure 3 and their discussion in the text. PhET Explorations: Circuit Construction Kit (DC Only), Virtual LabStimulate a neuron and monitor what happens. Pause, rewind, and move forward in time in order to observe the ions as they move across the neuron membrane. Section Summary
Conceptual QuestionsExercise 1Why should you not connect an ammeter directly across a voltage source as shown in Figure 9? (Note that script E in the figure stands for emf.) Figure 9 Exercise 2Suppose you are using a multimeter (one designed to measure a range of voltages, currents, and resistances) to measure current in a circuit and you inadvertently leave it in a voltmeter mode. What effect will the meter have on the circuit? What would happen if you were measuring voltage but accidentally put the meter in the ammeter mode? Exercise 3Specify the points to which you could connect a voltmeter to measure the following potential differences in Figure 10: (a) the potential difference of the voltage source; (b) the potential difference across ${R}_{1}$; (c) across ${R}_{2}$; (d) across ${R}_{3}$; (e) across ${R}_{2}$ and ${R}_{3}$. Note that there may be more than one answer to each part. Figure 10 Exercise 4To measure currents in Figure 10, you would replace a wire between two points with an ammeter. Specify the points between which you would place an ammeter to measure the following: (a) the total current; (b) the current flowing through ${R}_{1}$; (c) through ${R}_{2}$; (d) through ${R}_{3}$. Note that there may be more than one answer to each part. Problem ExercisesExercise 1What is the sensitivity of the galvanometer (that is, what current gives a fullscale deflection) inside a voltmeter that has a $1\text{.}\text{00}\text{M}\Omega $ resistance on its 30.0V scale? Show/Hide Solution Solution$\text{30}\phantom{\rule{0.25em}{0ex}}\mathrm{\mu A}$ Exercise 2What is the sensitivity of the galvanometer (that is, what current gives a fullscale deflection) inside a voltmeter that has a $\text{25}\text{.}0\text{k}\Omega $ resistance on its 100V scale? Exercise 3Find the resistance that must be placed in series with a $\text{25}\text{.}\mathrm{0\Omega}$ galvanometer having a $\text{50.0}\mathrm{\mu A}$ sensitivity (the same as the one discussed in the text) to allow it to be used as a voltmeter with a 0.100V fullscale reading. Show/Hide Solution Solution$1\text{.}\text{98 k}\Omega $ Exercise 4Find the resistance that must be placed in series with a $\text{25}\text{.}\mathrm{0\Omega}$ galvanometer having a $\text{50}\text{.}\mathrm{0\mu A}$ sensitivity (the same as the one discussed in the text) to allow it to be used as a voltmeter with a 3000V fullscale reading. Include a circuit diagram with your solution. Exercise 5Find the resistance that must be placed in parallel with a $\text{25}\text{.}\mathrm{0\Omega}$ galvanometer having a $\text{50}\text{.}\mathrm{0\mu A}$ sensitivity (the same as the one discussed in the text) to allow it to be used as an ammeter with a 10.0A fullscale reading. Include a circuit diagram with your solution. Show/Hide Solution Solution$$1\text{.}\text{25}\times {\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\Omega $$Exercise 6Find the resistance that must be placed in parallel with a $\text{25}\text{.}\mathrm{0\Omega}$ galvanometer having a $\text{50}\text{.}\mathrm{0\mu A}$ sensitivity (the same as the one discussed in the text) to allow it to be used as an ammeter with a 300mA fullscale reading. Exercise 7Find the resistance that must be placed in series with a $\text{10}\text{.}\mathrm{0\Omega}$ galvanometer having a $\text{100\mu A}$ sensitivity to allow it to be used as a voltmeter with: (a) a 300V fullscale reading, and (b) a 0.300V fullscale reading. Show/Hide Solution Solution(a) $3\text{.}\text{00 M}\Omega $ (b) $2\text{.}\text{99 k}\Omega $ Exercise 8Find the resistance that must be placed in parallel with a $\text{10}\text{.}\mathrm{0\Omega}$ galvanometer having a $\text{100\mu A}$ sensitivity to allow it to be used as an ammeter with: (a) a 20.0A fullscale reading, and (b) a 100mA fullscale reading. Exercise 9Suppose you measure the terminal voltage of a 1.585V alkaline cell having an internal resistance of $0\text{.}\text{100}\phantom{\rule{0.25em}{0ex}}\Omega $ by placing a $1\text{.}\text{00}\text{k}\Omega $ voltmeter across its terminals. (See Figure 11.) (a) What current flows? (b) Find the terminal voltage. (c) To see how close the measured terminal voltage is to the emf, calculate their ratio. Figure 11 Show/Hide Solution Solution(a) 1.58 mA (b) 1.5848 V (need four digits to see the difference) (c) 0.99990 (need five digits to see the difference from unity) Exercise 10Suppose you measure the terminal voltage of a 3.200V lithium cell having an internal resistance of $5\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\Omega $ by placing a $1\text{.}\text{00}\text{k}\Omega $ voltmeter across its terminals. (a) What current flows? (b) Find the terminal voltage. (c) To see how close the measured terminal voltage is to the emf, calculate their ratio. Exercise 11A certain ammeter has a resistance of $5\text{.}\text{00}\times {\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\Omega $ on its 3.00A scale and contains a $\text{10}\text{.}\mathrm{0\Omega}$ galvanometer. What is the sensitivity of the galvanometer? Show/Hide Solution Solution$\text{15}\text{.}\mathrm{0\; \mu A}$ Exercise 12A $1\text{.}\text{00}\text{M\Omega}$ voltmeter is placed in parallel with a $\text{75}\text{.}0\text{k}\Omega $ resistor in a circuit. (a) Draw a circuit diagram of the connection. (b) What is the resistance of the combination? (c) If the voltage across the combination is kept the same as it was across the $\text{75}\text{.}0\text{k}\Omega $ resistor alone, what is the percent increase in current? (d) If the current through the combination is kept the same as it was through the $\text{75}\text{.}0\text{k}\Omega $ resistor alone, what is the percentage decrease in voltage? (e) Are the changes found in parts (c) and (d) significant? Discuss. Exercise 13A $0\text{.}\text{0200\Omega}$ ammeter is placed in series with a $\text{10}\text{.}\text{00\Omega}$ resistor in a circuit. (a) Draw a circuit diagram of the connection. (b) Calculate the resistance of the combination. (c) If the voltage is kept the same across the combination as it was through the $\text{10}\text{.}\text{00\Omega}$ resistor alone, what is the percent decrease in current? (d) If the current is kept the same through the combination as it was through the $\text{10}\text{.}\text{00\Omega}$ resistor alone, what is the percent increase in voltage? (e) Are the changes found in parts (c) and (d) significant? Discuss. Show/Hide Solution Solution(a) Figure 12 (b) $10\text{.}\text{02}\phantom{\rule{0.25em}{0ex}}\Omega $ (c) 0.9980, or a $2.0\times {10}^{\mathrm{\u20131}}$ percent decrease (d) 1.002, or a $2.0\times {10}^{\mathrm{\u20131}}$ percent increase (e) Not significant. Exercise 14Unreasonable Results Suppose you have a $\text{40}\text{.}\mathrm{0\Omega}$ galvanometer with a $\text{25}\text{.}\mathrm{0\mu A}$ sensitivity. (a) What resistance would you put in series with it to allow it to be used as a voltmeter that has a fullscale deflection for 0.500 mV? (b) What is unreasonable about this result? (c) Which assumptions are responsible? Exercise 15Unreasonable Results (a) What resistance would you put in parallel with a $\text{40}\text{.}\mathrm{0\Omega}$ galvanometer having a $\text{25.}\mathrm{0\mu A}$ sensitivity to allow it to be used as an ammeter that has a fullscale deflection for $\text{10}\text{.}\mathrm{0\mu A}$? (b) What is unreasonable about this result? (c) Which assumptions are responsible? Show/Hide Solution Solution(a) $\text{66}\text{.}7\phantom{\rule{0.25em}{0ex}}\Omega $ (b) You can’t have negative resistance. (c) It is unreasonable that ${I}_{\mathrm{G}}$ is greater than ${I}_{\text{tot}}$ (see Figure 5). You cannot achieve a fullscale deflection using a current less than the sensitivity of the galvanometer.
