The Ammeter and the Voltmeter — Question 7

Question 7

Two finite-input readings determine an otherwise inaccessible open-circuit voltage

The same stable source is measured separately with two voltmeters. A 6.00 kilohm instrument reads 6.00 V; a 12.0 kilohm instrument reads 8.00 V. Model the source as emf E in series with positive resistance r.

1. Use the two loaded readings to infer E and r.

2. Predict the reading of an 18.0 kilohm voltmeter and the source short-circuit current within this linear model.

Question 7 - Solution

Two finite-input readings determine an otherwise inaccessible open-circuit voltage

(a) Method and calculation

$$6=E\frac6{r+6},\quad8=E\frac{12}{r+12}\quad\text{(resistances in kilohms)}.$$$$E=r+6=\tfrac23r+8\Rightarrow r=6.00\,\mathrm{k\Omega},\quad E=12.0\,\mathrm V.$$

(b) Interpretation and further calculation

$$V_{18}=12\frac{18}{6+18}=9.00\,\mathrm V.$$$$I_{\mathrm{sc}}=E/r=12/6000=2.00\,\mathrm{mA}.$$

Physical check

Different input resistances provide enough information to separate loading from the open-circuit voltage.