Lab E3: The Wheatstone Bridge

Introduction

The Wheatstone bridge is a circuit used to compare an unknown resistance with a known resistance. A schematic is shown below:

 

The unknown resistor is Rx, the resistor Rk is known, and the two resistors R1 and R2 have a known ratio , although their individual values may not be known. A galvanometer G measures the voltage difference VAB between points A and B. Either the known resistor Rk or the ratio is adjusted until the voltage difference VAB is zero and no current flows through G. When VAB = 0, the bridge is said to be "balanced".

 

 

Since VAB = 0, the voltage drop from C to A must equal the voltage drop from C to B, VCA = VCB. Likewise, we must have VAD = VBD. So we can write,

(1)           

(2)            .

Dividing (2) by (1), we have

(3)             .

Thus, the unknown resistance Rx can be computed from the known resistance Rk and the known ratio . Notice that the computed Rx does not depend on the voltage Vo; hence, Vo does not have to be very stable or well-known. Another advantage of the Wheatstone bridge is that, because it uses a null measurement, (VAB = 0), the galvanometer does not have to be calibrated.

In practice, the Wheatstone bridge is seldom used merely to determine the value of a resistor in the manner just described. Instead, it is usually used to measure small changes in Rx due, for instance, to temperature changes or the motion of microscopic defects in the resistor. As an example, suppose Rx = 106 W and we wanted to measured a change in Rx of 1W , resulting from a small temperature change. There is no ohmmeter which can reliably measure a change in resistance of 1 part in a million. However, the bridge can be set up so that VAB = 0 when Rx is exactly 106 W . Then any change in Rx , ,would result in a non-zero VAB, which, as we show below, is proportional to .

You would not weigh a cat by weighing a boat with and without a cat on board. Likewise, you would not want to measure very small changes in Rx by measuring Rx with and without the change. Instead, you want to arrange things so that the change in Rx, , is the entire signal. The bridge serves to "balance out" the signal due Rx, leaving only the signal due to .

To show that ,we consider Fig.1, and note that . We assume that the galvanometer is a perfect voltmeter, so that no current flows through it, even when the bridge is not balanced. We also assume that the bridge has been balanced with the sample resistance at an initial value of Rxo , so that . Then we consider what happens to VAB when the sample resistor is changed by a small amount to a new value .

Applying Kirchhoff's Voltage Law and Ohm's Law to the upper and lower arms of the bridge, we have

(4)           .

We are trying to find VAB, which we can relate to Ia and Ib.

(5)            .

We can use (4) to solve for Ia and Ib and then substitute for Ia and Ib in (5),

(6)            .

This equation shows how VAB depends on Rx. Notice that eqn.(6) yields when . To see how much VAB changes when Rx changes from Rxo to , we write

(7)       .

We wrote eqn.(7) by regarding VAB as a function of Rx, and remembering (from Calculus) that if then .

Substituting into (7) yields

(8)  

Finally, we remember that VAB,initial = 0, so the change in VAB is the same as VAB, and we have

(9)            .

 

Experiment

We will use a slide-wire Wheatstone bridge, in which the two resistors R1 and R2 are two portions of a single, uniform Ni-Cr wire. Electrical contact is made at some point along the wire by a sliding contact (this contact corresponds to point A). The two portions of the wire on either side of the contact have resistances R1 and R2, and the ratio is the same as the ratio of the lengths of the two portions of wire, . The lengths are readily measured with a meter stick which the wire rests upon.

Fig. 2. Physical Layout of the Wheatstone Bridge.

The 10W resistor in series with the 6V battery serves to limit the current through the bridge to less than 1A. (Higher currents might over-heat components of the bridge.) The rechargeable battery lasts only 6 hours when connected to the bridge, so please remember to disconnect it when done. The known resistor Rk is adjustable and can be set to any value from 1W to 999W in 1W steps with a decade resistance box, which is accurate to 0.02%. The Ni-Cr wire has a total resistance of about 2W . The sliding contact is spring loaded; you have to push it down to make contact to the wire at point A. There are two buttons on the sliding contact; push one or the other, but not both, to contact the wire.

The galvanometer has three spring-loaded buttons, labeled 1, 2, and 3. The buttons close internal switches which connect the galvanometer to the external voltage VAB . The 3 buttons correspond to 3 different sensitivities with button 1 the least sensitive and 3 the most sensitive. Always begin your measurements with button 1 depressed, and adjust the position of the sliding contact to balance the bridge (zero deflection on the galvanometer). Then re-balance the bridge with button 2 and then finally with button 3. Never begin with button 3. The galvanometer is quite sensitive and can be easily damaged if a large voltage is applied with button 3 depressed.

For Part 1 of the lab, the unknown resistor Rx is one of 5 coils of wire, numbered 1 through 5, mounted on a board. The lengths of the wires, their composition, and their gauge number are printed on the board. The gauge number is a measure of the thickness of the wire. 22 gauge wire has a diameter of 0.644mm; 28 gauge wire has a diameter of 0.321mm. For part 2 of the lab, Rx is a stand-alone coil of copper.

The resistance R of a wire is related to its length L, its cross-sectional area A, and its resistivity r by the formula

(10)           .

The resistivity r of a material depends on composition, on defect structure, and on the temperature. For metals, r is approximately constant at very low temperatures and increases approximately linearly with temperature (measured in 0K) at high temperatures.

At T = 20C, the resistivity of pure, defect-free, copper is . (RT stands for room temperature.)

 

Procedure

Part 1: Resistivity of Copper

In this section, you will use the bridge to make precise measurements of the resistance of each of the 5 coils of wire on the coil board. First, however, use the digital multimeter (DMM) to make an approximate measurement of the resistance of each coil. Also use the DMM to see how the resistance of the decade box changes when you turn the knobs. You must temporarily disconnect the coil board and the decade box from the bridge when testing them with the DMM. With the DMM, measure the resistance of each of the five coils to the nearest 0.1W . The resistance of the wire leads used to connect the DMM to the coils is not negligible, so first use the DMM to measure the resistance of the two wire leads in series, and then subtract this lead resistance from your measurements.

Now use the bridge to measure the resistance of each of the 5 coils. Before connecting the battery to the bridge, carefully check that all the connections are correct. Select a coil, attach it to the bridge, and set the decade box resistance Rk to be as near to Rx as possible(you know Rx roughly from your DMM measurements). Balance the bridge by moving the sliding contact along the wire while watching the galvanometer. Remember that when using the galvanometer you should always begin with button 1, then button 2, and finally button 3. With the bridge balanced, measure L1 and L2, and compute

(11)              .

Repeat this procedure for the other 4 coils.

For each of these coils, compute the resistivity, using your measured resistances, the data printed on the coil board, and eq'n (10). Make a table with the headings: coil #, R from DMM, R from bridge, and computed resistivity. Four of the 5 sample coils are made of copper (Cu). For these four coils, compute the average resistivity, and the uncertainty of the average (). Compare your average value with the known value. Coil #5 is made of a copper-nickel alloy. What is the ratio ?

Part 2. Temperature dependence of resistivity.

For this part, the unknown Rx is a copper wire coil (the one which is not attached to the coil board). We will use eq'n(9) to measure the change in resistance of the coil when its temperature is changed by plunging it into an ice bath. Note that eq'n(9) can be written as

(9')           

Prepare an ice bath by filling a beaker with ice from the icemaker (next to the video area) and adding some water. Measure room temperature and the ice bath temperature with the digital thermometer.

Measure the coil resistance Rx with the DMM and set Rk equal to Rx, as nearly as possible. With Rx and Rk connected in the bridge, measure with the DMM. See Fig. 2 to locate VCD. Note that VCD is not the battery voltage.

For this bridge measurement, we must replace the galvanometer in the bridge circuit with the DMM, set to DC volts with the most sensitive range(400mV). We need to do this because the galvanometer does not behave as an ideal voltmeter, as assumed in the derivation of eq'n (9).

With the sample coil at room temperature, balance the bridge and compute the coil's resistance at room temperature. Now without changing the position of the slide wire contact, place the coil into the ice bath and watch as the voltage VAB on the DMM changes. Record VAB after it reaches equilibrium. Use eq'n(9) to compute , the change in resistance of the coil.

As a measure of how sensitive the resistivity is to temperature changes, we can compute the fractional change in the resistivity divided by the change in temperature,

(12)           .

This quantity, multiplied by 100%, is the % change per degree. From your measurements, compute the % change in resistivity per degree for copper, and compare this value with the value you expect if . [Hint: If , then , where C is some constant, and .]

 

Questions

1. Two copper wires, labeled A and B, are at the same temperature, but the temperature is unknown. Wire B is twice as long and has half the diameter of wire A. Compute the ratio RB/RA.

2. A 28 gauge copper wire is 30 meters long. What is its resistance at room temperature?

3. Name two advantages of a Wheatstone bridge over an ordinary ohmmeter.

4. What is the formula for the total resistance of two resistors in parallel? Consider two resistors and . What is the total resistance of these two resistors in parallel. Give your answer to the nearest ohm.

 

5. Consider Fig.2 and recall that the total resistance of the Ni-Cr wire is 2W . If Rk and Rx are both very, very large compared to 2W , how much current flows through the Ni-Cr wire?

6. Suppose an unknown resistance Rx is measured with the bridge circuit shown in Fig. 2 and the result is Rx = 8.65W . The 6V battery is then replaced with an 10V battery and Rx is re-measured. What is the new measured value of Rx?

7. A sample of copper wire is 100m long and has a diameter of 0.250 mm. It's resistance is determined to be 38.0W . What is the resistivity of the copper in this wire?

8. (Counts as 3 questions) A Wheatstone Bridge such as in Fig. 1 has a . With Rk = 7.00W and the unknown resistor Rx at room temperature, the bridge is balanced with L1 = 44.0cm and . What is the value of Rx? While still attached to the bridge, the unknown Rx is placed in an oven, which raises its temperature by 23.0o C. The value of VAB is then found to be 0.087V. What is the resistance change of the unknown, D Rx? What is the % change per degree of the resistivity of this sample?