WHEATSTONE BRIDGE Objectives The Wheatstone bridge is a circuit designed to measure an unknown resistance by comparison with other known resistances. A slide-wire form of the Wheatstone bridge will be used to accomplish the following objectives: 1. Demonstration of the standard color code used to specify the value of commercially available resistors. 2. Explanation of the principles on which the operation of the Wheatstone bridge is based. 3. Determination of the value of several unknown resistances. Equipment List Theory 1. Slide-wire Wheatstone bridge 2. Standard decade resistance box 1000 3. Galvanometer 4. Switch with 10-kΩ resistor in parallel 5. Direct-current power supply 6. Five color-coded resistors range, 100 to 1000 to serve as unknowns 7. Assortment of spade, banana plug, and alligator clip leads 8. Multimeter with resistance scale Consider a circuit containing three resistors whose values are both known and adjustable, an unknown resistance, a power supply, and a galvanometer connected as shown in Figure 1. Current I from the power supply arrives at junction J, where it divides. The current in R 1 is I 1, and the current in R 3 is I 2,, where I = I 1 + I 2. By experimentally varying the values of the resistances, a condition can be achieved in which there is no current in the galvanometer G. This condition is called the balance condition. When there is no current in the galvanometer the current in R 1 has no place to go except through R 2 ; thus the current in R 2 is also equal to I 1. Similarly, when the balance condition holds, the current through R 4 must be the same as the current through R 3, and is thus equal to I 2.
Figure 1 Wheatstone bridge circuit. When there is no current in the galvanometer there is no potential difference between points A and B; in other words, points A and B are at the same potential. Thus, the change in potential from point J to point B V JB ) is equal to the potential change from point J to point A V JA ), and it follows that V JA = I 1 R 1 = V JB = I 2 R 3 and thus I 1 R 1 = I 2 R 3 1) Similarly, the potential change across R 2 V AK ) is the same as the potential change across R 4 V BK ), so that V AK = I 1 R 2 = V BK = I 2 R 4 and thus I 1 R 2 = I 2 R 4 2) If equation 1 is divided by equation 2, the currents cancel, and it follows that Therefore, when the balance condition has been experimentally achieved, equation 3 can be used to determine the value of an unknown resistance if three of the four values of resistance are known. In this experiment, a slide-wire form of the Wheatstone bridge, which is shown in Figure 2 will be used. In it, resistances R 3 and R 4 are replaced by a uniform wire between the points J and K with a sliding contact key at point B. Since the wire has a uniform cross section, the resistance of the two portions of wire JB and BK are proportional to their lengths. The ratio of their lengths, JB/BK, is equal to the ratio of their resistances R 3 /R 4. If R 1 is an unknown resistance R U, and R 2 is a known variable resistor R K, equation 3 becomes 4) 3)
The 10-kΩ resistor and switch S in series with the galvanometer are designed to protect the galvanometer. Be sure that you are aware of their proper function as described in the procedure before completing the connection of the power supply to the circuit. Figure 2 Slidewire form of the Wheatstone bridge. The value of resistors routinely used in electronic instrumentation are coded by a series of colored bands on the resistor. The key to the resistor color-coding system is given in Figure 3. Figure 3
Resistance is typically coded by a series of colored bands on the resistor. The key is given above in Figure 5. The four bands are placed with three equally spaced bands close to one end of the resistor followed by a space, and then a fourth band. The first two bands are the first two digits in the value of the resistor, and the third band gives the exponent of the power of 10 to be multiplied by the first two digits. This a resistor with its first three bands labeled Yellow-Violet-Red has a value of 47 x 10 2 Ω. Experimental Procedure 1. Using the resistor code table, read the nominal values of the five unknown resistors and record them in Data Table 1. Record the smallest value as #1, and then the remaining ones in increasing order. 2. Adjust the power supply voltage to 2.50 V. Leave the power supply fixed at this value for all the measurements. All measurements should be made with this same voltage, which has been chosen so that the currents in all resistors of the circuit will be small. This ensures that there is no heating of the resistors. Any significant heating of the resistors could cause differential increases in resistance, which would lead to errors. It would be ideal to use a smaller voltage, but our power supply will not go below 2.3 2.5 V. 3. Place the first unknown resistor in the Wheatstone bridge circuit in the position of R U in the circuit shown in Figure 2. Place the resistance box in the position of R K in Figure 2 and choose a value for R K approximately equal to the nominal value that you read from the resistor code for this unknown resistor. Record the value of R K in Data Table 1. Note that the value of R K should be given to one place beyond the last digit of the resistance box under the assumption that the uncertainty is not in the last digit set on the box, but in the digit beyond that. For example, if a resistance is set on the resistance box as 153 Ω, it should be recorded as 153.0 Ω because there is clearly not one unit of uncertainty in the 3.) 4. The 10-kΩ resistor and witch S in series with the galvanometer are designed to protect the galvanometer from excessive current. Be sure that each attempt to find a balance condition starts with switch S open. This places the resistor in series with the galvanometer and limits the current. 5. With switch S open, move the sliding contact at B until a balance is achieved i.e., zero current in the galvanometer. This is coarse balance. 6. With the system at course balance, close switch S to achieve maximum sensitivity and make a fine adjustment to achieve the balance condition. Because the galvanometer may have a small zero offset, determine the point where there is no deflection of the meter. This may not be at the zero of the meter. Record in Data Table 1 the values of JB and BK, the length of the two sections of wire at balance. Note that the Wheatstone bridge has a scale of 1 mm as the smallest marked division. Therefore, measurements of JB and BK should be made to the nearest 0.1 mm.
7. Using the same unknown resistor, repeat steps 3 through 6 above with two other values of R K, one value approximately 10% greater than the original R K, and one value approximately 10% less than the original R K. 8. Repeat steps 3 through 7 above for each of the other four unknown resistors. 9. Using the resistance scale on a multimeter, measure the value of each of the five unknown resistors and record those values in Data Table 2. Calculations 1. Using equation 4, calculate and record the three measured values for each of the five unknown resistors in the Calculations Table 1. 2. Calculate and record the mean and standard error for the three measurements of each of the five resistors. 3. Using the known values as the values from the table, calculate the percent error of each unknown resistor. Record these in Calculations Table 2.
Data Table 1 Calculations Table 1 #1 #2 #3 #4 #5 R k JB cm) BK cm) R U R U R U Data Table 2 # 1 2 3 4 5 Resistance Ω Ω Ω Ω Ω Calculations Table 2 # 1 2 3 4 5 Percent error