21-6. Null Measurements
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 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.
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 , 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.
Figure 2(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
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 .
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?
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.
Why can a null measurement be more accurate than one using standard voltmeters and ammeters? What factors limit the accuracy of null measurements?
If a potentiometer is used to measure cell emfs on the order of a few volts, why is it most accurate for the standard to be the same order of magnitude and the resistances to be in the range of a few ohms?
What is the of a cell being measured in a potentiometer, if the standard cell’s emf is 12.0 V and the potentiometer balances for and ?
Calculate the of a dry cell for which a potentiometer is balanced when , while an alkaline standard cell with an emf of 1.600 V requires to balance the potentiometer.
When an unknown resistance is placed in a Wheatstone bridge, it is possible to balance the bridge by adjusting to be . What is if ?
To what value must you adjust to balance a Wheatstone bridge, if the unknown resistance is , is , and is ?
(a) What is the unknown in a potentiometer that balances when is , and balances when is for a standard 3.000-V emf? (b) The same is placed in the same potentiometer, which now balances when is for a standard emf of 3.100 V. At what resistance will the potentiometer balance?
(a) 2.00 V
Suppose you want to measure resistances in the range from to using a Wheatstone bridge that has . Over what range should be adjustable?