# Learning Objectives

### Learning Objectives

By the end of this section, you will be able to do the following:

• Explain why a voltmeter must be connected in parallel with the circuit
• Draw a diagram showing an ammeter correctly connected in a circuit
• Describe how a galvanometer can be used as either a voltmeter or an ammeter
• 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
• 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 Figure 4.29). The internal construction of the simplest of these meters, and how they are connected to the system they monitor gives further insight into applications of series and parallel connections.

Figure 4.29 The fuel and temperature gauges (far right and far left, respectively) in this 1996 car are voltmeters that register the voltage output of sender units, which are hopefully proportional to the amount of gasoline in the tank and the engine temperature. (Christian Giersing)

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 4.30, 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 4.31, where the ammeter is represented by the symbol A).

Figure 4.30 (a) To measure potential differences in this series circuit, the voltmeter (V) is placed in parallel with the voltage source or either of the resistors. Note that terminal voltage is measured between points a and b. It is not possible to connect the voltmeter directly across the emf without including its internal resistance, $r.r.$ (b) A digital voltmeter in use. (Messtechniker, Wikimedia Commons)
Figure 4.31 An ammeter (A) is placed in series to measure current. All of the current in this circuit flows through the meter. The ammeter would have the same reading if located between points d and e or between points f and a as it does in the position shown. (Note that the script capital E stands for emf and $rr$ that stands for the internal resistance of the source of potential difference.)

# Analog Meters: Galvanometers

### 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 handheld calculator. The heart of most analog meters is a device called a galvanometer, denoted by G. Current flow through a galvanometer, $IG,IG, size 12{I rSub { size 8{G} } } {}$ produces a proportional needle deflection. This deflection is due to the force of a magnetic field on 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 $50 μA50 μA$ has a maximum deflection of its needle when $50 μA50 μA$ flows through it, reads half-scale when $25 μA25 μA size 12{2"5 "μA} {}$ flows through it, and so on.

If such a galvanometer has a $25-Ω25-Ω size 12{2"5-" %OMEGA } {}$ resistance, then a voltage of only $V=IR=50 μA25 Ω=1.25 mVV=IR=50 μA25 Ω=1.25 mV size 12{V= ital "IR"= left ("50" μA right ) left ("25" %OMEGA right )=1 "." "25"" mV"} {}$ produces a full-scale reading. By connecting resistors to this galvanometer in different ways, you can use it as either a voltmeter or an ammeter that can measure a broad range of voltages or currents.

#### Galvanometer as Voltmeter

Figure 4.32 shows how a galvanometer can be used as a voltmeter by connecting it in series with a large resistance, $R.R.$ The value of the resistance $RR size 12{R} {}$ 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 $25-Ω25-Ω size 12{2"5-" %OMEGA } {}$ galvanometer with a $50-μA50-μA$ sensitivity. Then 10 V applied to the meter must produce a current of $50 μA.50 μA. size 12{"50" μA} {}$ The total resistance must be

4.68 $Rtot=R+r=VI=10 V50 μA=200 kΩ, orRtot=R+r=VI=10 V50 μA=200 kΩ, or size 12{R rSub { size 8{"tot"} } =R+r= { {V} over {I} } = { {"10"" V"} over {"50" μA} } ="200"" k" %OMEGA } {}$
4.69 $R=Rtot−r=200 kΩ−25Ω≈200 kΩ.R=Rtot−r=200 kΩ−25Ω≈200 kΩ. size 12{R=R rSub { size 8{"tot"} } -r="200"k %OMEGA -"25" %OMEGA »"200"" k" %OMEGA } {}$

$RR size 12{R} {}$ is so large that the galvanometer resistance, $r,r,$ is nearly negligible. Note that 5 V applied to this voltmeter produces a half-scale deflection by producing a $25-μA25-μA size 12{2"5-"μA} {}$ current through the meter, 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.

Figure 4.32 A large resistance $RR$ placed in series with a galvanometer G produces a voltmeter, the full-scale deflection of which depends on the choice of $R.R. size 12{R} {}$ The larger the voltage to be measured, the larger $RR size 12{R} {}$ must be. (Note that $rr$ represents the internal resistance of the galvanometer.)

#### Galvanometer as Ammeter

The same galvanometer can also be made into an ammeter by placing it in parallel with a small resistance $R,R, size 12{R} {}$ often called the shunt resistance, as shown in Figure 4.33. 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, that an ammeter is needed that gives a full-scale deflection for 1.0 A and contains the same $25-Ω25-Ω size 12{2"5-" %OMEGA } {}$ galvanometer with its $50-μA50-μA size 12{"50"-μA} {}$ sensitivity. Since $RR size 12{R} {}$ and $rr size 12{r} {}$ are in parallel, the voltage across them is the same.

These $IRIR size 12{ ital "IR"} {}$ drops are $IR=IGrIR=IGr size 12{ ital "IR"=I rSub { size 8{G} } r} {}$ so that $IR=IGI=Rr.IR=IGI=Rr. size 12{ ital "IR"= { {I rSub { size 8{G} } } over {I} } = { {R} over {r} } } {}$ Solving for $RR size 12{R} {}$ and noting that $IGIG size 12{I rSub { size 8{G} } } {}$ is $50 μA50 μA size 12{"50" μA} {}$ and $II size 12{I} {}$ is 0.999950 A, we have

4.70 $R=rIGI=(25Ω)50 μA0.999950 A=1.25×10−3Ω.R=rIGI=(25Ω)50 μA0.999950 A=1.25×10−3Ω. size 12{R=r { {I rSub { size 8{G} } } over {I} } = $$"25" %OMEGA$$ { {"50" mA} over {0 "." "999950 A"} } =1 "." "25"´"10" rSup { size 8{-3} } %OMEGA } {}$
Figure 4.33 A small shunt resistance $RR size 12{R} {}$ placed in parallel with a galvanometer G produces an ammeter, the full-scale deflection of which depends on the choice of $R.R. size 12{R} {}$ The larger the current to be measured, the smaller $RR size 12{R} {}$ must be. Most of the current ($II$) flowing through the meter is shunted through $RR size 12{R} {}$ to protect the galvanometer. (Note that $rr$ represents the internal resistance of the galvanometer.) Ammeters may also have multiple scales for greater flexibility in application. The various scales are achieved by switching various shunt resistances in parallel with the galvanometer—the greater the maximum current to be measured, the smaller the shunt resistance must be.

# Taking Measurements Alters the Circuit

### 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, so the circuit is not appreciably affected (see Figure 4.34(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 4.34(b)). The voltage across the device is not the same as when the voltmeter is out of the circuit.

Figure 4.34 (a) A voltmeter having a resistance much larger than the device ($RVoltmeter>>RRVoltmeter>>R size 12{V">>"R} {}$) with which it is in parallel produces a parallel resistance essentially the same as the device and does not appreciably affect the circuit being measured. (b) Here the voltmeter has the same resistance as the device ($RVoltmeter≅RRVoltmeter≅R size 12{V simeq R} {}$) so that the parallel resistance is half of what it is when the voltmeter is not connected. This is an example of a significant alteration of the circuit and is to be avoided.

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, so the extra resistance is negligible (see Figure 4.35(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 4.35(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.

Figure 4.35 (a) An ammeter normally has such a small resistance that the total series resistance in the branch being measured is not appreciably increased. The circuit is essentially unaltered compared with when the ammeter is absent. (b) Here the ammeter’s resistance is the same as that of the branch so that the total resistance is doubled and the current is half what it is without the ammeter. This significant alteration of the circuit is to be avoided.

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 Knowledge

Making 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 solid-state electronics and null measurements can attain accuracies of one part in $106.106. size 12{"10" rSup { size 8{6} } } {}$