Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » College Physics » Null Measurements
Content endorsed by: OpenStax College

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Endorsed by Endorsed (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
  • OpenStax College

    This collection is included in aLens by: OpenStax College

    Click the "OpenStax College" link to see all content they endorse.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Pierpont C & TC display tagshide tags

    This module is included inLens: Pierpont Community & Technical College's Lens
    By: Pierpont Community & Technical CollegeAs a part of collection: "College Physics -- HLCA 1104"

    Click the "Pierpont C & TC" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Featured Content display tagshide tags

    This collection is included inLens: Connexions Featured Content
    By: Connexions

    Comments:

    "This introductory, algebra-based, two-semester college physics book is grounded with real-world examples, illustrations, and explanations to help students grasp key, fundamental physics concepts. […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Null Measurements

Module by: OpenStax College. E-mail the author

Summary:

  • Explain why a null measurement device is more accurate than a standard voltmeter or ammeter.
  • 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 Figure 1. (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 VV size 12{V} {}, which is related to the emf of the battery by V=emfIrV=emfIr size 12{V="emf" - ital "Ir"} {}, where II size 12{I} {} is the current that flows and rr size 12{r} {} is the internal resistance of the battery.

The emf could be accurately calculated if rr size 12{r} {} were very accurately known, but it is usually not. If the current II size 12{I} {} could be made zero, then V=emfV=emf size 12{V="emf"} {}, and so emf could be directly measured. However, standard voltmeters need a current to operate; thus, another technique is needed.

Figure 1: An analog voltmeter attached to a battery draws a small but nonzero current and measures a terminal voltage that differs from the emf of the battery. (Note that the script capital E symbolizes electromotive force, or emf.) Since the internal resistance of the battery is not known precisely, it is not possible to calculate the emf precisely.
The diagram shows equivalence between two circuits. The first circuit has a cell of e m f script E and an internal resistance r connected across a voltmeter. The equivalent circuit on the right shows the same cell of e m f script E and an internal resistance r connected across a series combination of a galvanometer with an internal resistance r sub G and high resistance R. The currents in the two circuits are shown to be equal.

A potentiometer is a null measurement device for measuring potentials (voltages). (See Figure 2.) A voltage source is connected to a resistor R,R, say, a long wire, and passes a constant current through it. There is a steady drop in potential (an IRIR size 12{ ital "IR"} {} drop) along the wire, so that a variable potential can be obtained by making contact at varying locations along the wire.

Figure 2(b) shows an unknown emfxemfx size 12{"emf" rSub { size 8{x} } } {} (represented by script ExEx size 12{"emf" rSub { size 8{x} } } {} in the figure) connected in series with a galvanometer. Note that emfxemfx size 12{"emf" rSub { size 8{x} } } {} 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, emfx=IRxemfx=IRx size 12{"emf" rSub { size 8{x} } = ital "IR" rSub { size 8{x} } } {}, where RxRx size 12{R rSub { size 8{x} } } {} 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 emfxemfx size 12{"emf" rSub { size 8{x} } } {} is directly sensed.

Now, a very precisely known standard emfsemfs size 12{"emf" rSub { size 8{s} } } {} is substituted for emfxemfx size 12{"emf" rSub { size 8{x} } } {}, and the contact point is adjusted until the galvanometer again reads zero, so that emfs=IRsemfs=IRs size 12{"emf" rSub { size 8{s} } = ital "IR" rSub { size 8{s} } } {}. In both cases, no current passes through the galvanometer, and so the current II size 12{I} {} through the long wire is the same. Upon taking the ratio emfxemfs emfxemfs size 12{ { {"emf" rSub { size 8{x} } } over {"emf" rSub { size 8{s} } } } } {}, II size 12{I} {} cancels, giving

emfxemfs=IRxIRs=RxRs.emfxemfs=IRxIRs=RxRs. size 12{ { {"emf" rSub { size 8{x} } } over {"emf" rSub { size 8{s} } } } = { { ital "IR" rSub { size 8{x} } } over { ital "IR" rSub { size 8{s} } } } = { {R rSub { size 8{x} } } over {R rSub { size 8{s} } } } } {}
(1)

Solving for emfxemfx size 12{"emf" rSub { size 8{x} } } {} gives

emfx=emfsRxRs.emfx=emfsRxRs. size 12{"emf" rSub { size 8{x} } ="emf" rSub { size 8{s} } { {R rSub { size 8{x} } } over {R rSub { size 8{s} } } } } {}
(2)
Figure 2: The potentiometer, a null measurement device. (a) A voltage source connected to a long wire resistor passes a constant current II size 12{I} {} through it. (b) An unknown emf (labeled script ExEx in the figure) is connected as shown, and the point of contact along RR size 12{R} {} is adjusted until the galvanometer reads zero. The segment of wire has a resistance RxRx size 12{R rSub { size 8{x} } } {} and script Ex=IRxEx=IRx size 12{E rSub { size 8{x} } = ital "IR" rSub { size 8{x} } } {}, where II size 12{I} {} is unaffected by the connection since no current flows through the galvanometer. The unknown emf is thus proportional to the resistance of the wire segment.
Two circuits are shown. The first circuit has a cell of e m f script E and internal resistance r connected in series to a resistor R. The second diagram shows the same circuit with the addition of a galvanometer and unknown voltage source connected with a variable contact that can be adjusted up and down the length of the resistor R.

Because a long uniform wire is used for RR size 12{R} {}, the ratio of resistances Rx/RsRx/Rs size 12{R rSub { size 8{x} } /R rSub { size 8{s} } } {} 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 emfxemfx size 12{"emf" rSub { size 8{x} } } {} 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 Rx/RsRx/Rs size 12{R rSub { size 8{x} } /R rSub { size 8{s} } } {} and in the standard emfsemfs size 12{"emf" rSub { size 8{s} } } {}. Furthermore, it is not possible to tell when the galvanometer reads exactly zero, which introduces error into both RxRx size 12{R rSub { size 8{x} } } {} and RsRs size 12{R rSub { size 8{s} } } {}, and may also affect the current II size 12{I} {}.

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 Figure 3. Such configurations are limited in accuracy, because the meters alter both the voltage applied to the resistor and the current that flows through it.

Figure 3: Two methods for measuring resistance with standard meters. (a) Assuming a known voltage for the source, an ammeter measures current, and resistance is calculated as R=VIR=VI size 12{R= { {V} over {I} } } {}. (b) Since the terminal voltage VV size 12{V} {} varies with current, it is better to measure it. VV size 12{V} {} is most accurately known when II size 12{I} {} is small, but II size 12{I} {} itself is most accurately known when it is large.
The diagram shows two circuits. The first one has a cell of e m f script E and internal resistance r connected in series to an ammeter A and a resistor R. The second circuit is the same as the first, but in addition there is a voltmeter connected across the voltage source E.

The Wheatstone bridge is a null measurement device for calculating resistance by balancing potential drops in a circuit. (See Figure 4.) 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 R1R1 size 12{R rSub { size 8{1} } } {} and R2R2 size 12{R rSub { size 8{2} } } {} are precisely known, while the arrow through R3R3 size 12{R rSub { size 8{3} } } {} indicates that it is a variable resistance. The value of R3R3 size 12{R rSub { size 8{3} } } {} can be precisely read. With the unknown resistance RxRx size 12{R rSub { size 8{x} } } {} in the circuit, R3R3 size 12{R rSub { size 8{3} } } {} 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 IRIR size 12{ ital "IR"} {} drops along abc and adc are the same. Since b and d are at the same potential, the IRIR size 12{ ital "IR"} {} drop along ad must equal the IRIR size 12{ ital "IR"} {} drop along ab. Thus,

I1R1=I2R3.I1R1=I2R3. size 12{I rSub { size 8{1} } R rSub { size 8{1} } =I rSub { size 8{2} } R rSub { size 8{3} } } {}
(3)

Again, since b and d are at the same potential, the IRIR size 12{ ital "IR"} {} drop along dc must equal the IRIR size 12{ ital "IR"} {} drop along bc. Thus,

I1R2=I2Rx.I1R2=I2Rx. size 12{I rSub { size 8{1} } R rSub { size 8{2} } =I rSub { size 8{2} } R rSub { size 8{x} } } {}
(4)

Taking the ratio of these last two expressions gives

I1R1I1R2=I2R3I2Rx.I1R1I1R2=I2R3I2Rx. size 12{ { {I rSub { size 8{1} } R rSub { size 8{1} } } over {I rSub { size 8{1} } R rSub { size 8{2} } } } = { {I rSub { size 8{2} } R rSub { size 8{3} } } over {I rSub { size 8{2} } R rSub { size 8{x} } } } } {}
(5)

Canceling the currents and solving for Rx yields

Rx=R3R2R1.Rx=R3R2R1. size 12{R rSub { size 8{x} } =R rSub { size 8{3} } { {R rSub { size 8{2} } } over {R rSub { size 8{1} } } } } {}
(6)
Figure 4: The Wheatstone bridge is used to calculate unknown resistances. The variable resistance R3R3 size 12{R rSub { size 8{3} } } {} is adjusted until the galvanometer reads zero with the switch closed. This simplifies the circuit, allowing RxRx size 12{R rSub { size 8{x} } } {} to be calculated based on the IRIR size 12{ ital "IR"} {} drops as discussed in the text.
This complex circuit diagram shows a galvanometer connected in the center arm of a Wheatstone bridge arrangement. All the other four arms have a resistor. The bridge is connected to a cell of e m f script E and internal resistance r.

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 R1R1 size 12{R rSub { size 8{1} } } {}, R2R2 size 12{R rSub { size 8{2} } } {}, and R3R3 size 12{R rSub { size 8{3} } } {}, which contribute to the uncertainty in RxRx size 12{R rSub { size 8{x} } } {}.

Check Your Understanding

Identify other factors that might limit the accuracy of null measurements. Would the use of a digital device that is more sensitive than a galvanometer improve the accuracy of null measurements?

Solution

One factor would be resistance in the wires and connections in a null measurement. These are impossible to make zero, and they can change over time. Another factor would be temperature variations in resistance, which can be reduced but not completely eliminated by choice of material. Digital devices sensitive to smaller currents than analog devices do improve the accuracy of null measurements because they allow you to get the current closer to zero.

Section Summary

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

Conceptual questions

Exercise 1

Why can a null measurement be more accurate than one using standard voltmeters and ammeters? What factors limit the accuracy of null measurements?

Exercise 2

If a potentiometer is used to measure cell emfs on the order of a few volts, why is it most accurate for the standard emfsemfs size 12{"emf" rSub { size 8{s} } } {} to be the same order of magnitude and the resistances to be in the range of a few ohms?

Problem Exercises

Exercise 1

What is the emfxemfx size 12{"emf" rSub { size 8{x} } } {} of a cell being measured in a potentiometer, if the standard cell’s emf is 12.0 V and the potentiometer balances for Rx=5.000ΩRx=5.000Ω size 12{R rSub { size 8{x} } =5 "." "000" %OMEGA } {} and Rs=2.500ΩRs=2.500Ω size 12{R rSub { size 8{s} } =2 "." "500" %OMEGA } {}?

Solution

24.0 V

Exercise 2

Calculate the emfxemfx size 12{"emf" rSub { size 8{x} } } {} of a dry cell for which a potentiometer is balanced when Rx=1.200ΩRx=1.200Ω size 12{R rSub { size 8{x} } =1 "." "200" %OMEGA } {}, while an alkaline standard cell with an emf of 1.600 V requires Rs=1.247ΩRs=1.247Ω size 12{R rSub { size 8{s} } =1 "." "247" %OMEGA } {} to balance the potentiometer.

Exercise 3

When an unknown resistance RxRx size 12{R rSub { size 8{x} } } {} is placed in a Wheatstone bridge, it is possible to balance the bridge by adjusting R3R3 size 12{R rSub { size 8{3} } } {} to be 2500Ω2500Ω size 12{"2500" %OMEGA } {}. What is RxRx size 12{R rSub { size 8{x} } } {} if R2R1=0.625R2R1=0.625 size 12{ { {R rSub { size 8{2} } } over {R rSub { size 8{1} } } } =0 "." "625"} {}?

Solution

1 . 56 k Ω 1 . 56 k Ω size 12{1 "." "56 k" %OMEGA } {}

Exercise 4

To what value must you adjust R3R3 size 12{R rSub { size 8{3} } } {} to balance a Wheatstone bridge, if the unknown resistance RxRx size 12{R rSub { size 8{x} } } {} is 100Ω100Ω size 12{"100" %OMEGA } {}, R1R1 size 12{R rSub { size 8{1} } } {} is 50.0Ω50.0Ω size 12{"50" "." 0 %OMEGA } {}, and R2R2 size 12{R rSub { size 8{2} } } {} is 175Ω175Ω size 12{"175" %OMEGA } {}?

Exercise 5

(a) What is the unknown emfxemfx size 12{"emf" rSub { size 8{x} } } {} in a potentiometer that balances when RxRx size 12{R rSub { size 8{x} } } {} is 10.0Ω10.0Ω size 12{"10" "." 0 %OMEGA } {}, and balances when RsRs size 12{R rSub { size 8{s} } } {} is 15.0Ω15.0Ω size 12{"15" "." 0 %OMEGA } {} for a standard 3.000-V emf? (b) The same emfxemfx size 12{"emf" rSub { size 8{x} } } {} is placed in the same potentiometer, which now balances when RsRs size 12{R rSub { size 8{s} } } {} is 15.0Ω15.0Ω size 12{"15" "." 0 %OMEGA } {} for a standard emf of 3.100 V. At what resistance RxRx size 12{R rSub { size 8{x} } } {} will the potentiometer balance?

Solution

(a) 2.00 V

(b) 9.68 Ω9.68 Ω size 12{9 "." "68 " %OMEGA } {}

Exercise 6

Suppose you want to measure resistances in the range from 10.0Ω10.0Ω size 12{"10" "." 0 %OMEGA } {} to 10.0 kΩ10.0 kΩ size 12{"10" "." 0" k" %OMEGA } {} using a Wheatstone bridge that has R2R1=2.000R2R1=2.000 size 12{ { {R rSub { size 8{2} } } over {R rSub { size 8{1} } } } =2 "." "000"} {}. Over what range should R3R3 size 12{R rSub { size 8{3} } } {} be adjustable?

Solution

Range = 5 . 00 Ω to 5 . 00 k Ω Range = 5 . 00 Ω to 5 . 00 k Ω size 12{"Range=5" "." "00 " %OMEGA " to "5 "." "00"" k" %OMEGA } {}
(7)

Glossary

null measurements:
methods of measuring current and voltage more accurately by balancing the circuit so that no current flows through the measurement device
potentiometer:
a null measurement device for measuring potentials (voltages)
ohmmeter:
an instrument that applies a voltage to a resistance, measures the current, calculates the resistance using Ohm’s law, and provides a readout of this calculated resistance
bridge device:
a device that forms a bridge between two branches of a circuit; some bridge devices are used to make null measurements in circuits
Wheatstone bridge:
a null measurement device for calculating resistance by balancing potential drops in a circuit

Collection Navigation

Content actions

Download module as:

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks