The Ammeter and the Voltmeter — Question 5

Question 5

A voltmeter across the specimen makes the ammeter include a second current

A voltmeter of resistance RV=10.0 kilohm is connected across an unknown resistor R. An upstream ideal ammeter reads the sum of specimen and voltmeter currents. The readings are V=4.00 V and IA=5.00 mA.

1. Correct the naive ratio V/IA to obtain R.

2. Derive the general fractional underestimate in the naive resistance and identify when this arrangement is accurate.

Question 5 - Solution

A voltmeter across the specimen makes the ammeter include a second current

(a) Method and calculation

$$I_V=V/R_V=.400\,\mathrm{mA},\quad I_R=I_A-I_V=4.60\,\mathrm{mA}.$$$$R=\frac V{I_A-V/R_V}=\frac{20000}{23}\,\Omega\simeq869.6\,\Omega.$$

(b) Interpretation and further calculation

$$R_{\mathrm{naive}}=V/I_A=R\parallel R_V,\quad\frac{R-R_{\mathrm{naive}}}R=\frac R{R+R_V}.$$$$R\ll R_V\Rightarrow R_{\mathrm{naive}}\simeq R.$$

Physical check

The ammeter current belongs to the entire parallel combination, not to the specimen alone.