Consider two resistors, R1 (1 kΩ) and R2 (3 kΩ), connected in parallel (Fig. 1a). According to Faraday's Law, a time-varying magnetic field H, increasing linearly with time, induces a constant 1 mA in the loop. What is the voltage across nodes A-B? According to Ohm's Law, the voltages across the two resistors should differ. But how is that possible if both resistors are connected to the same nodes A and B?
The answer may surprise you. The measured voltage depends on the position of the wires connecting it to the voltmeter! If two identical voltmeters are joined across nodes A-B (Fig. 1b), the simultaneous measured voltages will be −1 V and +3 V.
Faraday's Law governs the measured voltage, as explained in a paper by professor Robert H. Romer.1 The voltage is defined as the line integral of electric field from node A to B along a path C:
The induced electric field in our example is nonconservative, because the closed-loop integral of the electric field is nonzero:
Thus, the voltage in Equation 1 is path-dependent (in our example).
Integrating the electric field from node A to B along path C1 (Fig. 1b, again) gives a different value than when integrating along path C2. Thus, the measured voltage depends on which path the voltmeter "sees." Another way to understand this action is to consider current in the loop, which is 1 mA through both resistors. A given voltmeter sees a voltage drop across either the 1-kΩ or the 3-kΩ resistor.
To observe this phenomenon, set up an experiment as shown in Figure 2. The magnetic field is produced by a ferrite-core inductor (solenoid) driven by a sinewave. Neither the solenoid size, sinewave amplitude and frequency, nor resistor values are critical. But the resulting induced voltage must be large enough for measurement.
Using a two-channel oscilloscope, measure the voltages simultaneously with a probe across R1 from the left and a second probe across R2 from the right. As expected for the values shown, the sinewave amplitude induced across the 3-kΩ resistor is three times larger than that across the 1-kΩ resistor, and of opposite polarity (Fig. 3).
Because the value of measured voltage depends on how it's measured, you can arrange the voltmeter and its connecting wires to serve as a position sensor. Unlike optical position sensors whose narrow sensing field requires a precise initial position to ensure detection, the sensor of Figure 2 requires only that the detector be anywhere in the left (or right) hemisphere.
U1 and associated components form a simple amplitude detector that compares the sinewave amplitude with a dc voltage generated by the divider R3/R4. When sensing on the left (1 kΩ), the amplitude is too small to turn on the comparator, so its output is logic low. When sensing on the right (3 kΩ), the amplitude is large enough to cause oscillation at the comparator output. That signal is rectified to produce a logic high output.
1Romer, Robert H., "What do 'Voltmeters' Measure? Faraday's Law in a Multiply-Connected Region," American Journal of Physics, Vol. 50, No. 12 (Dec. 1982), p. 1089-1093.