Experiment 4
Resistance and Resistivity
A slide wire Wheatstone bridge is used to determine the resistance of coils of wire made from different materials. If the lengths and the diameters of the coils of wire are also known, then the resistivity of the material from which the wire is made can be determined.
Theory
Bridge circuits are designed to allow the determination of the value of an unknown circuit element such as a resistor, capacitor, or an inductor. The circuit diagram for a typical bridge is shown in Figure 1.
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The bridge elements are connected between junctions AC, BC, AD, and BD. V represents either an AC or DC voltage source and G represents a null detecting device such as a galvanometer, a voltmeter, or an oscilloscope.
Generally, one or more of the circuit elements in the bridge can be varied until the potential difference between junctions C and D (VCD) is zero. When this situation exists, the bridge is said to be balanced or is "nulled." The following relationships then hold for the voltages in the main branches:
Figure 1. A typical bridge circuit.
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(1)
and
(2)
When (1) is divided by (2) and rearranged, the voltage across any branch can be found in terms of the voltages across the remaining three. For example, the voltage between Junctions A and D is
(3)
The Wheatstone bridge is shown schematically in Figure 2. The coils of wire whose resistance is to be determined is connected between junctions A and D, and a known value of resistance is connected between B and D. A potentiometer or a long wire is connected between A and B with a tap at point C. The position of the tap can be altered, thereby, changing the resistances, RAC and RBC, on either side of point C. These changes then vary the voltage VAC and VBC.
Figure 2. Schematic of the Wheatstone bridge.
When the bridge is in the null condition (galvanometer reads zero),equation (3) holds. Thena current flows from A to D to B(IADB, and another currentflows from A to C to B(IACB). Knowing that the voltage across a resistor is IR (Ohm's Law), equation (3) can be expressed as
(4)
The value of RACis proportional to the length of wire LAC, and the value of RBCis proportional to (LAB–LAC).In this experiment we are using a 100 cm long wire as a potentiometer so that LAB=100. IfRAD is the unknown resistance, and RBD is the known resistance box value.
Equation (4) can now be written as
(5)
This is the working equation for the Wheatstone bridge.
The resistivity of the coil of wire with resistance Rx can now be determined. For a wire with a uniform cross-sectional area, the resistance is
.
When this expression is rearranged,
, (6)
where is the resistivity, lx isthe length of the coil of wire, and A is its cross-sectional area. When the cross-sectional area is expressed in terms of its diameter, d, the expression for its resistivity becomes
. (7)
The diameters of the unknown wires are given by the “gauge no.”: G# 22=6.439 x 10-4 m; G# 26=4.049 x 10-4 m;G# 28=3.211 x 10-4 m.
Apparatus
o wire slide apparatuso coil of copper wire
o DC power supplyo coil of German silver wire
o galvanometero coil of Nichrome wire
o standard resistance box, ±0.2%o 5 leads
Procedure
1)Connect the Wheatstone bridge circuit with the standard resistance box between junctions B and D and the coil of copper wire of unknown resistance between junctions A and D. Record the gauge number of the wire coil and its length.
2)Turn on the DC power supply and adjust the voltage or current knob to some convenient value.
3)Set the tap atjunction C to the 50 cm.
4)Depress KEY 1, 5 VOLTS button on the galvanometer. Pull and replace the plugs on the resistance box until the galvanometer needle is zeroed or nearly zeroed. Now depress KEY 2, 0.1 VOLTS button and again pull and replace the plugs until the needle is zeroed or nearly zeroed. Finally, depress the most sensitive button (KEY 3-GALV) and again attempt to zero the reading by adjusting the resistance box. CAUTION: If the most sensitive button (KEY 3-GALV) is depressed first, too much current may flow through the galvanometer causing it to bum out.
5)Hold the most sensitive button (KEY 3-GALV) depressed and slide the tap until the galvanometer is zeroed. If uncertainties are, to be calculated, then find the range of values of L that will zero the galvanometer. Record these values as Lminand Lmax.
6)Repeat steps (1) through (5) using the coil of German (nickel) silver wire. Repeat again with the Nichrome (Chromel) wire.
Analysis
Use (5) and (7) to determine the resistivity of copper, German silver, and Nichrome.
If uncertainties are to be calculated, then the value of LAC in (5) is expressed as
,
and its uncertainty is
.
An estimate of the reliability of the resistivity values can be found using the “fractional uncertainty” method. Assuming that the length and cross-section of the wire are known to great accuracy, the uncertainty in the resistivity is due to the fractional uncertainties in Rx.The uncertainty therefore is given by
, (8)
here the uncertainties in the diameter and length of the wire on the coil are neglected. In (8), L = and(Note that the standard resistance box has a tolerance of ±0.2%. If a different resistance box is used, then the fractional uncertainty of the resistance will be different.)
In a results table, report the experimental values of the resistivity for the three materials, their uncertainties (if calculated), the book values of the resistivities, and percentage errors when appropriate. Accepted resistivity values are: copper=1.72 x 10-8ohm-m; nickel silver=33 x 10-8ohm-m;copper= 100 x 10-8ohm-m.
Error Analysis
Describe the factor(s) that you felt caused the greatest amount of error in your determination of the resistivity.Be specific and indicate how each error would have affected your results.
Conclusion and Questions; some of these should be answered in the theory section, some in the error analysis, some in the conclusion statement.
- Was the experiment a success? Did you measure the accepted resistivity value to the level of uncertainty of your measurement?
- On what microscopic process does the resistivity depend?
- Does the resistivity of a conductor increase, decrease, or remain the same when the temperature of the conductor increases? Explain. What about a semiconductor? Why?
- Show in your theory section that you can derive the basic equations of this experiment: (4), (7) and (8). Make sure to define your variables, use schematics, and explain your steps. Do not merely copy the theory section in the handout, instead summarize it in your own style.
- Explain the problems associated with getting a reliable value from the resistance box.
- Explain why L= (100-L) in the uncertainty formula.
- The last two terms inside the brackets in (8) give a measure of the fractional uncertainty associated with the slide wire potentiometer used in this experiment. Suppose that the uncertainty in the wire reading,L,is ±0.05 cm. If LAC is 50.00 cm, what is the total fractional uncertainty of the last two terms in the uncertainty expression? If LAC were10.00, then what is the total fractional uncertaintyof the last two terms in the uncertainty expression? Does this provide a clue as to why the slide wire reading, L, should be as close as possible to the 50 cm position?
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