Question 2.23: In a certain commercial Wheatstone bridge, R1 is a fixed 1-k...
In a certain commercial Wheatstone bridge, R_1 is a fixed 1-kΩ resistor, R_3 can be adjusted in 1-Ω steps from 0 to 1100 Ω, and R_2 can be selected to be 1 kΩ, 10 kΩ, 100 kΩ, or 1 MΩ. a. Suppose that the bridge is balanced with R_3 = 732 Ω and R_2 = 10 kΩ. What is the value of R_x? b. What is the largest value of R_x for which the bridge can be balanced? c. Suppose that R_2 = 1 MΩ. What is the increment between values of R_x for which the bridge can be precisely balanced?
1. From Equation 2.91, we have
R_x = \frac{R_2}{R_1}R_3 (2.91)
R_x = \frac{R_2}{R_1}R_3=\frac{10 kΩ}{1 kΩ} × 732 Ω = 7320 Ω
Notice that R_2/R_1 is a scale factor that can be set at 1, 10, 100, or 1000, depending on the value selected for R_2. The unknown resistance is the scale factor times the value of R_3 needed to balance the bridge.
2. The maximum resistance for which the bridge can be balanced is determined by the largest values available for R_2 and R_3. Thus,
R_{x max} = \frac{R_{2 max}}{R_1}R_{3 max}=\frac{1 MΩ}{1 kΩ} × 1100 Ω = 1.1 MΩ
3. The increment between values of R_x for which the bridge can be precisely balanced is the scale factor times the increment in R_3:
R_{xinc} = \frac{R_2}{R_1}R_{3inc}=\frac{1 MΩ}{1 kΩ} × 1 Ω = 1 kΩ