C.16:8.3.1 - A voltmeter changes the voltage it reads

The sought quantity is the open-circuit voltage E of a source. Model the source as an ideal voltage E in series with resistance R_s; the connected voltmeter has input resistance R_m. The meter closes the circuit. Ohm’s law gives current I=E/(R_s+R_m), and the indication is V=I R_m. The measurement relation is therefore E=V(1+R_s/R_m).

For E=10 V and R_s=R_m=1 megohm, the indication is 5 V. The known resistance ratio recovers the open-circuit value as 10 V. The difference comes from the measurement interaction.

If both E and R_s are unknown, one indication leaves several pairs compatible with it. For V=5 V, R_m=1 megohm and an available bound 0.8≤R_s≤1.2 megohm, the conditional voltage interval is 9≤E≤11 V. That interval may answer the question. When a narrower answer is needed, a second indication with a different known input resistance supplies another equation, provided the source stays unchanged and the circuit model still applies. Repeating the original arrangement supplies the same relation and leaves this ambiguity.

These calculations use an ideal circuit. For an obtained measurement result, include uncertainty in the indications and resistances and any model inadequacy that affects the use.