Physics

LABORATORY EXPERIMENT

The Wheatstone Bridge

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Date:___________________________

Experiment-TheWheatstoneBridge.doc Page 1

LABORATORY EXPERIMENT The Wheatstone Bridge Introduction The Wheatstone bridge, which was invented by Samuel Christie, is a circuit that is used to accurately measure unknown resistances. The circuit is named after the British born physicist, Sir Charles Wheatstone who used it extensively in research on Ohm’s law in the early 1840s. Different modifications of the Wheatstone bridge have wide application in electronic circuits. The bridge is used in conjunction with a galvanometer which consists of a coil of mounted wire that enables it to rotate freely on a pivot in a magnetic field provided by a permanent magnet. The galvanometer provides a means for gauging the balance condition of the Wheatstone bridge. The bridge circuit has technical limitations, however, if it is used to measure very high resistances. Aims (a) Today’s laboratory will introduce students to the Wheatstone bridge as a null

measuring system for determining an unknown resistance, using the balance condition. This produces a more accurate measure than an ohmmeter as it is not subject to the possible inaccuracies of a meter reading.

Reference Serway, R. A. and Beichner, R.J. (2000), Physics for Scientists and Engineers with Modern Physics, 5th ed., Saunders College Publishing, Ch. 28. Preliminary Questions You will need to answer the following question and attach your answers to this laboratory Script. The Wheatstone bridge circuit consists of four resistor arms, a power supply and a sensitive ammeter such as a galvanometer. The circuit is illustrated in figure 1 with the four resistors R1, R2, R3 and R4 forming the four arms. The power supply causes the currents indicated, I, I1, I 2, I3, I4, I5 to flow. The bridge is balanced when the potentials at points B and C are equal, i.e. VB = VC, hence, I5 = 0 which results in a reading on the galvanometer of zero.

Figure 1

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1. When the bridge is balanced VB = VC. Therefore, VAB = VAC and VBD = VCD

Using Kirchoff’s voltage law and Ohm’s law, prove that, at balance, (I5 = 0),



R1 R2

 R3 R4

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2. Suppose R4 is an unknown resistance and the bridge is balanced. Express R4 in terms of the other resistance values. _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ Now calculate R4 if R1 = 1000 Ω , R2 = 500 Ω and R3 = 3000 Ω. _________________________________________________________________ _________________________________________________________________ _________________________________________________________________

3. If ∆R4 is the uncertainty in R4, write down an expression for ∆R4 in terms of R1, R2 and R3 and their uncertainties, i.e. ∆R1, ∆R2 and ∆R3. _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________

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4. Show that if R1 and R2 have no uncertainty – or the uncertainty is negligible compared to that of R3 and R4 – then :



R4 R4

 R3 R3

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Therefore, show that if R4 = 10 kΩ and R3 = 1 kΩ then ∆R4 = 10∆R3.

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5. A transducer is a device which converts one quantity into another that is more readily

measured. For example, the loudspeaker is a transducer which converts electrical signals into pressure waves or sound. In today’s experiment, the thermistor is to be used as a transducer to change a temperature variation into a current variation. What is a thermistor? _________________________________________________________________ _________________________________________________________________

6. Does the resistance of a thermistor vary linearly or non-linearly with temperature? _________________________________________________________________

7. A reasonable estimate for the resistance of a thermistor – similar to that used in the thermistor experiment – at room temperature is 600 Ω – 1 kΩ. If this thermistor forms the fourth arm of a Wheatstone bridge with R1 = R2 = 1000 Ω, calculate the value of R3 that is required to balance the bridge at room temperature (20 °C). _________________________________________________________________ _________________________________________________________________

8. If the current flowing through a thermistor was to change, would the resistance of the thermistor alter? Why?

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Procedure Part 1: The Null Measuring Wheatstone Bridge In this experiment, the Wheatstone bridge resistors R1 and R2 are fixed, R3 is a variable resistance in the form of a calibrated decade box and R4 is the unknown resistance. The value of R4 can be determined by adjusting R3 to produce a balanced bridge. It is easy to adjust the ratio R1/R2 to a simple ratio such as 10:1, 1:1 or 1:10, which is why the special circuit board provided has fixed resistors (10 Ω to 1000 Ω) which are to be used as R1 and R2. The calibrated decade resistance box is to be used for R3. Before wiring the circuit, please note the following points: (a) Whenever you connect a resistance box to a circuit, ensure some of the dials

are off zero so that there is resistance in the circuit. (b) Have the circuit checked before you connect the power supply. Connect up the Wheatstone bridge using the special circuit board as illustrated in figure 1, paying attention to how components have been placed on the board. Use R1 = R2 = 1000 Ω, R3 = the decade box, R4 = the unknown resistance and adjust the power supply to 3.0 volts. Note: The galvanometer connections do not have a preferred polarity, as current can

flow through in either direction. When your circuit has been checked, balance the bridge by adjusting R3 until the galvanometer reads zero. Use the sensitivity control of the galvanometer as follows: (a) Set the control to its least sensitive position (fully anti-clockwise), then rotate

the control in a clockwise direction. As a balanced condition is reached, the sensitivity can be gradually increased.

(b) Normally, the actual value of R3 will only read when the galvanometer sensitivity is at a maximum.

(c) Before commencing the next measurement, turn the sensitivity to a minimum. Record your results in the following table, varying R1 and R2 as indicated. Estimate the uncertainty in R3 from which you will be able to deduce the uncertainty in R4, assuming that there is negligible uncertainty in R1 and R2. To estimate the uncertainty in R3, find the range of dial settings of R3 which give a deviation of one scale division (on the galvanometer) away from the true balance setting.

R1 (Ω)

R2 (Ω)

R3 ± ∆R3 (Ω)

(decade box)

R4 ± ∆R4 (Ω)

1000 1000

1000 100

100 1000

100 100

Table 1

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1. Comment on how well the range of measurements of R4 agree within the experimental uncertainties quoted in the table.

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2. State the value, complete with the uncertainty, of R4 from your measurement.

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3. If both R1 and R2 now have a 1 per cent tolerance i.e. ∆R/R = 1/100, how would that

alter your final result for R4?

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For accurate resistance measurements, it appears that the three known resistances, R1, R2 and R3 need to be known to 0.1% or better. It is useful to know that the uncertainty in R1 and R2 can be removed from the problem by simply interchanging R1 and R2 and repeating the measurements.

4. This raise the question of the best choice of resistances R1 and R2.

Inspection of Table 1 should indicate that a ratio of R2:R1 of 10:1 does not give 10 times the accuracy of R2:R1 = 1:1. Moreover, for a Wheatstone bridge, it can be shown that the greatest sensitivity in locating the balance point, using a galvanometer as the detector, occurs when all arms of the bridge have the same resistance. Therefore, when determining an unknown resistance, it is good practice to commence with R1 = R2 at some medium value, for example, 1 kΩ, obtain an approximate measurement for R4, for example, 620 Ω, and repeat the measurement with R1 = R2 = 600 Ω to find a more precise value for R4. Note: the balance condition does not involve the magnitude of the voltage V across the bridge. Therefore, if V is altered, it should not change the balance condition for the same resistance values. Set the supply to 6 V and repeat the initial measurement of R4 to verify the preceding point.

R3 ± ∆R3 (Ω) (decade box)

R4 ± ∆R4 (Ω)

R1 = R2 = 1000 Ω

Table 2

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5. Comment on this result compared with previous determinations of R4. Now repeat this test using a thermistor as the unknown resistance.

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With the thermistor as R4, use the Wheatstone bridge to measure its resistance at a supply voltage of 3 volts and then repeat for 6 volts. Once the circuit is connected, allow several minutes for the thermistor to reach equilibrium before balancing. It may be necessary to shield the thermistor from air currents by placing it in an empty beaker.

R3 ± ∆R3 (Ω)

(decade box)

R4 ± ∆R4 (Ω)

(thermistor)

V = 3 V, R1 = R2 = 1000 Ω

V = 6 V, R1 = R2 = 1000 Ω

Table 3

6. Is the above result expected? Try to explain any discrepancy between the results

obtained using a resistor and a thermistor.

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7. In the introduction to this experiment, it was stated that this method avoids the

limitations on accuracy due to reliance on meter readings. Does the use of galvanometer deflections invalidate this claim? Explain. _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ ________________________________________

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