Ohm's law and Kirchhoff's voltage law (KVL) provide powerful tools for conventional circuit analysis (mesh analysis), but if time-varying magnetic fields are present, Faraday's law must be invoked as well. The additional current induced by a time-varying magnetic field must be accounted for by adding a term to Ohm's law and KVL. This introduction of Faraday's law into the circuit-analysis equations produces unexpected anomalies: Two voltages appear to coexist simultaneously between two nodes in a circuit, and the voltages appear to depend on position of the voltmeter leads.
Analysis Without Magnetic Fields
To understand the influence of time-varying magnetic fields, we should first review the KVL and Ohm's law equations for a circuit with no magnetic field present. Then, in the next section we can compare them with the corresponding equations in which a time-varying magnetic field is present.
With no magnetic field present, one commonly uses KVL and Ohm's law to perform a circuit analysis based on the mesh technique. As commonly stated in textbooks, KVL says the algebraic sum of all voltages around a closed loop equals zero (Eq. 1):
(1) (KVL)
Consider the circuit of Fig. 1 with the magnetic field turned off. If you take a voltmeter and measure the voltages across each component in the loop, the sum of those voltages equals zero as predicted by KVL (Eq. 2). (For the counter-clockwise direction chosen, note that voltages across the resistors are negative.)
-VR1 - VR2 + U1 = 0 (2)
You can solve for the component values in Fig. 1 by applying Ohm's law and KVL. First, obtain an equation for loop current by substituting from equations 3 and 4 into equation 2. Then, solving for the current yields equation 5:
VR1 = IR1 (3)
VR2 = IR2 (4)
Note that KVL can be written in integral form. Textbooks on electromagnetic theory define voltage as the vector integral of the electric field (E) along a path (dl), such as that from node A to node C in Fig. 1 (Eq. 6). With the magnetic field off, the closed-loop integral from node A to C to E and back to A equals zero (Eq. 7). Thus, Kirchhoff's voltage law can be written in integral form: the closed-loop integral of the electric field is equal to zero.
(6)
(Bold font indicates vectors)
(7)
(KVL in integral form)
Analysis with Magnetic Fields
Now, turn on the magnetic field in Fig. 1. The field varies with time, which induces current in the loop, and that condition calls for the use of Faraday's law: The integral of the tangential component of electric-field intensity around a closed loop equals the time rate-of-change for magnetic flux passing through a surface bounded by that loop (Eq. 8):
(8)
(Faraday's Law)
where B is the magnetic field, A is the area of the surface in question, and Ф is the total magnetic flux through that area. The direction of the induced current depends on the direction of the magnetic field. In Fig. 1, the induced current is clock-wise as shown if the magnetic field is pointing out of the page. Total current is now the sum of current due to the battery (IU1) and that due to magnetic induction (IMAG):
I = IU1 + IMAG (9)
Ohm's law must be modified (extended) to account for the additional current:
V = IU1R + IMAGR (10)
(Extended Ohm's Law)
Kirchhoff's voltage law must be extended as well. A comparison of equations 1, 7 and 8 shows that KVL is extended by adding the -dФ/dt term to the right side of equation 1:
(11)
(Extended KVL)
Equations 2 through 5 can be rewritten to include the time-dependent magnetic field components:
VR1 = (IU1 + IMAG)R1 (13)
VR2 = (IU1 + IMAG)R2 (14)
Thus, equations 1 through 5 have been extended to account for current induced by the magnetic field, forming Equations 11 through 15. Equation 11 is the extended Kirchhoff voltage law and equation 15 is the extended Ohm's law, with the sign of the dФ/dt term indicating the direction of current. These equations look simple enough, but they seem to describe a paradox.
Using equations 12 through 15, consider an analysis of the Fig. 1 circuit with time-varying magnetic field. The voltage across U1 (nodes A and F) is VAF = U1. But, VAF also equals current in the loop times the two resistances:
VAF= U1 (16)
We now have two possible voltages across nodes A and F. In fact, there are two possible voltages for each pair of nodes that include a component in Fig. 1. See squations 16 through 25. For a simple comparison, we arbitrarily set U1= 2 V, dФ/dt = 1 V, R1 = 2 kΩ, and R2 = 4 kΩ. Then, the loop current according to equation 15 is 0.5 mA.
VAF = 2V (18)
VBC = I(R1) = 1V (20)
VBC = U1 - I(R2) = 0V (21)
VDE = I(R2) = 2V (22)
VDE = U1 - I(R1) = 1V (23)
VEF = I(0O) = 0V (24)
VEF = U1 - I(R1) - I(R2) = -1V (25)
Nodes B and C are especially interesting, because the current through R1 is non-zero, yet the voltage across it could be zero. Similarly, nodes E and F represent a short (zero-ohm) wire, yet the voltage across these nodes is non-zero. So which voltage is it? Nothing is wrong with the math. Two voltages do coexist simultaneously! Mathematically, the voltage you get depends on the path of integration taken in your measurement. Remember that voltage is a vector integral of electric field along a given path. If a time-dependent magnetic field is present, the integration is path dependent. In simpler terms, the voltage depends on how the measuring circuit (the voltmeter) is connected to the nodes.
As shown in Fig. 2, voltmeter #1 measures the voltage across nodes A and F from the left-hand side, obtaining a measurement of U1 = 2 V. In contrast, voltmeter #2 measures the voltage across nodes A and F (B and E are the same as A and F) from the right-hand side, with the result
Contrary to a common misconception, the induced voltage is distributed not in the wires connecting the resistors, but within the resistors. The integral of electric field inside a wire is zero, so the voltage across a wire is zero. By sliding the probe contact from point A to point B, lab experiments verify that the voltage drop across the connecting wires is zero. Thus, the voltage on voltmeter #1 does not change. Similarly, sliding the contact from F to E does not change the voltage on voltmeter #1. The same applies to voltmeter #2. Sliding the contact from B to A or from E to F does not change the reading. The voltmeter probes are arranged to minimize interference from the magnetic field.
The measured voltage appears to depend on the position of the probes. Voltmeter #1 acts like an electric field integrator that integrates the electric field inside the battery U1, and voltmeter #2 integrates the electric field inside R1 and R2. Different integration paths yield different voltages. The following case further demonstrates this position-dependent effect.
Consider Fig. 3, in which an audio signal (a 1-kHz sinewave) is attenuated by a volume-control potentiometer (R1) and fed to an audio amplifier, whose output is analyzed by a spectrum analyzer. A nearby electric motor creates magnetic interference in the loop formed by R1 and the audio-signal source. For simplicity, R1 has been replaced with a 1-kΩ and a 10-kΩ resistor connected in series, and the loop that catches the magnetic flux was enlarged to 1 in2. Two physical board layouts were tested (Fig. 4).
Figs. 5a and 5b show plots of the audio-amplifier output spectra. The 1-kHz audio test tone is identical in both cases, but the amplitude of 300-Hz motor interference depends only on the ground connection. The worst magnetic interference (-62 dBc) is based on Fig. 4a, in which the audio amplifier receives interference voltage from the 10-kΩ resistor (Fig. 5a). In other words, the audio amplifier acts as an electric field integrator that integrates the electric field inside the 10-kΩ resistor.
On the other hand, Fig. 5b shows the interference voltage received from the 1-kΩ resistor. The lesser interference shown in this plot (-78.5 dBc) represents an improvement of 16.5 dB. (The expected interference ratio is 20 dB, because the resistor ratio is 10-to-1, but the loading effect of the audio amplifier's input impedance lowers the interference amplitude in Fig. 5a.)
This phenomenon has been experimentally verified. Note that the voltage across two nodes is not clearly defined; it can be either of the two voltages, depending on how the wires are positioned. This experiment demonstrates that the voltage between two nodes is no longer a simple algebraic expression, but a vector integral of the electric field along a given path. Because the integral is path- or position-dependent, integrating along a different path yields a different voltage. Equations 9 through 15 in the previous section do not clearly predict the position-dependent effect, so they must be used very carefully.
References
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Romer, Robert H. “What do ‘Voltmeters’ Measure? Faraday's Law in a Multiply-Connected Region,” American Journal of Physics. Vol. 50, No. 12 (Dec. 1982), pp. 1089-1093.
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