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.
I agree with Peter Haak -- I would much prefer to see and download a PDF file rather than HTML. The new ED web site seems to be less user friendly and more difficult to navigate than the old one. The lack of PDF files is a perfect example of this. Please reinstate PDF versions ASAP!
EDITOR RESPONSE: Many of our Web visitors prefer the pdf format for printouts. That's why we have provided a printer friendly option for each editorial article available online in html format. This option will print both the main text and all the figures as a single multiple-page document.
Anthony Smith -June 03, 2003
Unfortunately also here only html instead of .pdf. More difficult to read. Takes more time for downloading. I have to select each design entry separately. And I never now how it is going to look after printing (think of conversion from US to European pages etc.). I would very much like to have the previous .pdf format !
Peter Haak -May 22, 2003
The previous massage from me has a shortcut: The ZERO in the lower equation has to be replaced by (I Rx ). Thank you.
Norbert Renz -May 21, 2003
I want to suggest that ED should review such articles before publishing one. The author is completely wrong and may confuse a lot of students and others. 1) Its wrong that the U(AB) depends from position of the Voltmeter. In this example U(AB)=1 Volt. The Voltmeter has to be connected correct between A-B. If placed outside the wires have to be feed out as koax, or wire plane parallel do B-Field so no extra voltage may be induced. 2) The calculation is wrong. The author lost Kirchhoff-2. dPHI/dt(AB)-R1 I = -dPHI/dt(BA)+R3 I where I=dPHI(BB)/dt / (R1+R2) = 4V / 4kOhm = 1mA Using the actual numbers:2V-1V = -2V+3V =1V !!! 3) The author suggest that the calculation has to figured out, so that his wrong measurement must result but shows no calculation. This is shown here. U(wl) - U(AB) - U(R1) = 0 = -U(wr) + U(BA) + U(R2) where U(wl) is the voltage induced in the wires of the voltmeter = U(AB)... So he calculates the voltage drop over the resistors but not U(AB)!! This is done in the real circuit be feed the meters leads along the wires to the resistors and have the same effect as a galvanic connection to the resistors itself independent where he places the solder points!!! 4) The setup to measure is critical because you may accidential get a configuration like Fig.1b where a area appears bn. Rx and the Voltmeter so a part of the B-Field is canceled out. The nature has warned the author because his measurement shows no exact result what he comments with the word >about<!? With best regards Norbert Renz WolfVision
Norbert Renz -May 20, 2003
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