Answer: For a resistance of 1 ohm, the error of measurement with circuit (a) is 0.1 ohm, or 10%; with circuit (b) it is 0.001 ohm, or 0.1%. For a resistance of 500 ohms the corresponding values are 0.1 ohm or 0.02% and 167 ohms or 33.4%.
Explanation:
If V and I are the voltmeter and ammeter readings, the calculated resistance R' = V/I is equal to the equivalent resistance of branch bb' with circuit (a) and the equivalent resistance of branch cc' with circuit (b). Hence R' is related to the resistance R, in the first case, by
R'2 = R + Ra
and, in the second case, by
R'2 = RRb/(R + Rb).
Comparing the calculated values of the resistance with the actual value of resistance R , we find the error made by using the circuits shown in the figure. These errors are connected with the fact that, in circuit (a), the calculation with the voltmeter reading does not take into account the drop in voltage across the internal resistance of the ammeter, and, in circuit (Z>), the calculation with the ammeter reading does not take into account the current flowing in the voltmeter. This is the reason why the resistance R' calculated from the instrument reading is larger than the true resistance R in the first case, and smaller in the second case.
With a decrease in the value of the measured resistance R in circuit (a), the voltage drop across the ammeter takes up an increasing share of the voltmeter reading, and the relative error resulting from the use of this circuit will increase.
In circuit (b), with a decrease in the resistance R, the current flowing through the voltmeter decreases. The error in the ammeter reading, and consequently, the relative error in the calculation will also decrease.
For small resistances it is preferable to use circuit (b) and for large resistances, circuit (a).
