We present a simple demonstration of the skin effect by observing the current distribution in a wide rectangular strip conductor driven at frequencies in the 0.25–5 kHz range. We measure the amplitude and phase of the current distribution as a function of the transverse position and find that they agree well with numerical simulations: The current hugs the edges of the strip conductor with a significant variation in phase across the width. The experimental setup is simple, uses standard undergraduate physics instructional laboratory equipment, and is easy to implement as a short inclass demonstration. Our study is motivated by modeling ac magnetic near fields in the vicinity of a rectangular trace on an atom chip.
I. INTRODUCTION
While direct current (dc) flows uniformly through a conductor, a timevarying or alternating current (ac) travels preferentially along the skin of a conductor. Interactions between the alternating current, the associated magnetic field, and the induced electric field create transverse spatial variations in both the current's amplitude and phase. This behavior defines the skin effect, which has been known since the late 19th century,^{1–10} and the current distribution has been characterized for various wire profiles. However, with the exception of a cylindrical conductor of circular cross section (i.e., round wire), numerical approaches^{11–24} and approximations^{25–32} are required to determine the current distribution.
Primary interest in the skin effect concerns the increase in ac resistance due to the effective decrease in the wire crosssectional area. For example, a 1 mm diameter copper wire with a 1 GHz ac increases its resistance per unit length to about 2.6 $ \Omega / m $, a factor of 120 compared to dc, while decreasing the selfinductance by a similar factor.^{33} At high frequencies, braided and Litz wire can help mitigate the skin effect, and printed circuit board designs must account for this effect. To this end, much of the research on this topic predicts and measures bulk observables, such as ac resistance as a function of frequency^{12,16,24,34} or wave penetration depth.^{35}
In contrast, our interest in the skin effect concerns the associated ac magnetic field in the vicinity of a ribbonlike wire. In our research on ac Zeeman forces, we manipulate ultracold atoms with radiofrequency (rf) magnetic nearfields generated by currents in the microfabricated 100 μm wide traces of an atom chip.^{36,37} While probing such μmscale rf fields is challenging,^{38} basic nearfield predictions involving the skin effect can be tested experimentally with lower frequencies at the mmscale.
In a ribbonlike conductor, with a rectangular cross section that is much wider than its thickness, the skin effect tends to concentrate the current along the two edges of the ribbon [see Fig. 1(a)] and is referred to as the lateral skin effect. Notably, the current density does not hug the edges of the ribbon as tightly as in a bulk conductor (the traditional skin depth) and does not vary appreciably over the thickness of the ribbon.
In this paper, we present a simple method for probing the current distribution and phase due to the skin effect in a ribbonlike conductor. An amplified pickup coil detects the current distribution in the conductor by measuring the amplitude and phase of the ac magnetic nearfield just above the conductor. Our experimental scheme requires only standard lab equipment and is sufficiently simple for implementation as a classroom demonstration or as an undergraduate lab exercise. Also, we compare our measurements with predictions from several numerical models of varying complexities and dimensionalities. Figure 1 shows the current density and magnetic nearfield predictions for four models, which largely agree with each other.
Our experimental method works best for ribbonlike conductors, which generate a onedimensional spatial variation of the current density, so long as the skin depth is larger than the conductor thickness. Our ribbon conductor dimensions fall in this lateral skin effect regime for all the frequencies that we consider. Our method is reminiscent of the one developed by Tsuboi and Kunisue^{39} for analyzing magnetic fields produced by large ac in thin conducting plates.^{40} Recent work has measured the skin effect in a stripline transmission line^{41} and in a rectangular conductor at high current.^{42,43} In contrast, Ampère's law and cylindrical symmetry guarantee that for a round wire, the external magnetic field is unaffected by the radial current distribution within it. In this case, direct measurements of the current density redistribution due to the skin effect must use an internal probe, such as neutrons,^{44} NMR,^{45} a liquid,^{46} or a segmented^{35} conductor, to name a few.
This paper is structured as follows: in Sec. II, we present the relevant electromagnetic theory and numerical approaches, followed by details of the experimental method in Sec. III. We present and compare our measurements of the current and magnetic field distributions with simulations in Sec. IV and conclude in Sec. V. The appendices provide additional details on the calibration procedure and also review analytic expressions for the skin effect and its phase distribution.
II. THEORY
The round wire admits the only known exact analytic solution to Eq. (1) in a finite volume.^{29,49} The round wire solution for the current density is given by $J(r)=C J 0 [(1\u2212i)r/\delta ]$, valid at all frequencies, where r is the radial coordinate, $ J 0 ( r ) $ is the Bessel function of the first kind, and C is a normalization constant. In the high frequency limit ( $ \delta \u226a R $, for radius R), this solution reduces to $J(r)\u2243 J max e \u2212 ( 1 + i ) ( R \u2212 r / \delta ) $ for $ r \u2243 R $, with J_{max} the current density at the edge of wire. Notably, this example shows that the phase of the current distribution also varies with r across the conductor, a fact that is often overlooked in discussions of the skin effect. Specifically, the phase wraps by $ 2 \pi $ for every δ of penetration into the wire as its amplitude decreases by $ 1 / e $. N.B.: at a given time, the current flow is not all in the same direction.
A ribbonlike conductor, with thickness 2T much smaller than the width 2W, is in the lateral skin effect regime for $ \delta \u226b T $.^{26} In this case, the current distribution falls off from the two ribbon edges to a finite value in the middle with a $ 1 / e $ characteristic decay length λ that is larger than the skin depth δ. While the fall off does not have a closed form, at low frequency ( $ \delta 2 > W T $), it is roughly polynomial, while at high frequency ( $ \delta 2 < W T $), it is more exponentiallike. In the very high frequency limit ( $ \delta 2 \u226a W T $), the lateral current distribution has an analytic form (see Appendix A), which is plotted in black in Fig. 1(b). Notably, the phase also varies across the conductor width but less so than in the round wire case (see Appendix B and Fig. 9).
We have provided a supplemental online animation,^{50} which illustrates the timeevolving nature of the current density's phase and amplitude across a wide range of frequencies. The lateral current density is shown to advance by rotating around the position axis in complex space [as in Fig. 2(e)]. The instantaneous amplitude is projected onto the real axis to show the measured value. Generally, the phase (and current) at the edges leads the current at the center of the strip. At very high frequencies, the total current lags 90° behind the neardc phase (and current).
A. Pedagogical explanation
We present a pedagogical explanation for the ac skin effect in Fig. 2 by expanding on an approach given by Zangwill.^{51} The reason that an ac hugs the skin of a conductor is because the driving current creates a magnetic field that is the largest at the edge, which, in turn, generates opposing eddy currents. The net result is a current distribution that is the largest in amplitude at the skin and out of phase with the source. To demonstrate further, we begin by considering a round wire (radius R) driven by a low enough frequency ac such that the skin effect is a small perturbation on the uniform current distribution (i.e., $ \delta \u226b R $). We can use the following steps illustrated in the indicated parts of Fig. 2 to calculate the first order correction to a uniform input current distribution J with a low frequency ω:

shows the sinusoidal time dependence of the uniform input ac, its associated magnetic field B, the first order contribution to the induced electric field $ \u2207 \u2192 \xd7 E \u2192 ind = \u2212 d B \u2192 / d t $, and the resulting eddy current $ \Delta J = \sigma E ind $.

shows the uniform input ac density $ J = J 0 \u2009 sin \u2009 ( \omega t ) $ and its inphase quasistatic magnetic field $ B = B 0 \u2009 sin \u2009 ( \omega t ) $. The field $ B 0 = \mu J 0 r / 2 $ increases linearly outwards from the center.

shows the spatial dependence of $ \u2212 d B / d t = \u2212 \omega B 0 \u2009 cos \u2009 ( \omega t ) $, which then generates an induced electric field E_{ind} (first order) via Faraday's law.

shows the eddy current distribution $ \Delta J $ generated by $ E ind = \mu J 0 \omega ( r 2 / 4 ) \u2009 cos \u2009 ( \omega t ) $ along the wire axis. Applying Ohm's law, we obtain the first order correction to the current density $ \Delta J = ( J 0 / 2 ) ( r / \delta ) 2 \u2009 cos \u2009 ( \omega t ) $, which increases quadratically from the wire center.

shows the resulting total current density J_{tot} from the quadrature sum of the input current J and the first order correction $ \Delta J $.
Thus, to first order, the ac skin effect results in a current density J_{tot} that varies radially in magnitude as $ 1 + ( r / \delta ) 4 / 8 $ and radially in phase as well. As the drive frequency ω is increased, higher order contributions in $ ( r / \delta ) 2 $ must be included. The physics is similar for a strip conductor; however, the computation of J_{tot} is more involved.
B. Simulations
We compute J(x) using four different methods and then calculate the corresponding xcomponent of the magnetic nearfield $ B x ( x , y h ) $, evaluated at the effective height y_{h} of our pickup coil. Models that consider the vertical extent of J show less than a part in 10^{3} variation vertically for our parameters, and this extent is averaged over for J(x) values.
We use two commercial electromagnetic solvers to compute J(x) and $ B ( x , y h ) $. FEKO uses a method of moments (MoM) approach to solve a finitelength model of our strip, giving the only longitudinal current description, but the current sheet model gives no vertical information. Flux uses a finite element method (FEM) to solve a 2D transverse cross section of the strip, without longitudinal information.^{52}
We have also directly implemented two numerical algorithms that model an infinite length strip in the transverse plane. The first algorithm is by Silvester^{15} and solves Eq. (2) by decomposing the strip cross section onto a Cartesian grid of square dxdy elements and calculating the mutual induction between them. The second algorithm by Belevitch et al.^{27} uses a flat line of current expanded in even polynomial powers to solve the same equation (see Appendix A for a summary of this method).
We find that the four numerical models give comparable results for the phase and amplitude of the current density J(x) and magnetic field B(x) (see Fig. 1). Three of the models give very similar results, but we find that the FEKO phase results deviate somewhat from the others and depend on the discretization mesh geometry.^{53} We note that the phase offsets of each model in Figs. 1(d) and 1(e) have been adjusted so that the phase over the center portion of the strip corresponds to 0°. In the case of FEKO, the small contribution to the ac magnetic field from the supply wires is subtracted out in Figs. 1(c) and 1(e).
III. EXPERIMENTAL METHOD
We measure the ac distribution J(x) (in A/m) laterally across a thin aluminum strip via the ac magnetic nearfield that it produces at the surface of the conductor. We use a labbuilt pickup coil located just above the strip to sample the ac magnetic field via the voltage induced in the coil.
The basic experimental setup [see Figs. 3(a) and 3(d)] consists of a thin strip of aluminum driven by a sinusoidal current source. The pickup coil is scanned transversely across the surface of the conducting strip, and its induced emf signal is sent to a batterypowered amplifier (gain $ \u223c 10 4 $ with 10 kHz bandwidth, based on two OP27E opamp gain stages). The output of the amplifier is then displayed on an oscilloscope, along with the signal from an isolated Hall sensor (LEM model HX 10NP) that monitors the total current through the strip. A series ammeter provides an additional rms measurement of the ac.
The aluminum alloy strip has width $ 2 W = $ 80.1(1) mm and thickness $ 2 T = $ 0.63(1) mm. We measure its dc conductivity to be $\sigma =2.50(6)\xd71 0 7 ( \Omega \u2009 m ) \u2212 1 $ with a fourpoint measurement. The aluminum strip is mounted on medium density fiberboard (MDF) with doublesided tape, and electrical connections soldered on washers are bolted to the strip with through holes at its two ends. The strip is about 0.9 m long but could be much shorter since the pickup coil measures very little variation in the signal along the strip's length, except at the ends.
We direct an ac with an amplitude of 1.85 A (1.3 A_{rms}) through the strip using a voltage controlled current source driven by the sinewave output of a function generator. At neardc frequencies, the current density is essentially uniform at J_{dc} = 1.85 A/8 cm $\u224823.1\u2009A/m$, which corresponds to a surface magnetic field of $ B d c \u2248 0.145 $ G. We use drive frequencies in the range of 0.25–5 kHz. Our current source (labbuilt, based on a LM675 opamp) operates up to 5 kHz, while below 250 Hz, the small pickup coil signal is too noisy.
The amplified pickup coil is very sensitive to environmental noise, such as rf communication signals (e.g., Bluetooth and WiFi) and the 60 Hz noise (and associated harmonics) emanating from nearby electrical devices. We found that for low noise measurements, the overhead fluorescent lights and cellphones had to be turned off while taking data. Alternatively, in a noisy environment, directing the batterypowered amplifier signal to a lockin amplifier could provide a cleaner signal.
Care was taken to route the ac supply wires away from the pickup coil to minimize crosstalk. We suspect that the placement of these supply wires on one side of the conductor may contribute to the slight asymmetry in the current distribution observed in Fig. 5.
A. Pickup coil
The pickup coil [Figs. 3(b) and 3(c)] consists of a machined, highly elongated rectangular copper loop of external dimensions of 114 mm × 1.85 mm with an inner gap measuring 110.6(1) mm × 0.38(5) mm, centered at a height of 0.80(5) mm. The base for construction was a 114 mm× 79 mm doublesided copperclad electronics prototyping circuit board (PCB). We initially used two loops on the front and back in series for higher sensitivity but switched to a single loop for improved spatial resolution. The PCB construction ensures that the pickup loop is flat in a plane and that the two coil planes are parallel.
We machined the PCB into the pickup coil using a desktop CNC milling machine (Carvey, Inventable Inc.). Bulk copper removal was done with a regular end mill bit ( $ 1 / 8 \u2033 $ fishtail upcut bit), while a specialized 0.1 mm diameter bit (P3.2501) was used for the regions directly adjacent to the wire loop and within it.
We note that alternative singleturn and multiturn pickup coils based on wrapping a thin wire around a plastic card were effective at producing a signal. However, the signal amplitude showed a significant asymmetry when the coils were rotated 180° around the vertical yaxis. The PCBbased coil minimizes this asymmetry.
B. Measurement theory
C. Calibration
We used a twopart calibration procedure. First, the frequencydependent gain of the coilamplifier system was examined using the pickup coil to measure the magnetic field near an aluminum rod of the circular cross section. For a known current in the rod, the drop in the signal at higher frequencies can be attributed to the bandwidth of the coilamplifier system, independent of the skin effect in the rod. Second, we measure the field above the rectangular strip at a low frequency (250 Hz), where the current is nearly evenly distributed. Averaging over the middle region of a known current density allows us to relate the amplified voltage V_{sig} to the known average current density and the surface magnetic field strength. Measurements of V_{sig} are divided by the linear ω scaling, corrected for frequency dependent amplifier gain, and multiplied by the voltagetoJ and voltagetoB factors to produce real values of J and B_{x} for the data in Figs. 5 and 6. The full calibration procedure is detailed in Appendix C.
Phases are measured using the time delay between the zero crossings of the Hall current sensor and V_{sig}. The phase is presented relative to the x = 0 center phase ( $\u22610\xb0$), since pickup ( $\u221290\xb0$), inversion ( $180\xb0$), and bandwidth (unique to frequency) were not studied with sufficient precision.
IV. RESULTS
The main results of this paper are shown in Figs. 5 and 7, where we plot the amplitude and phase, respectively, of the pickup coil signal vs transverse position x. The amplitude measurements in Fig. 5 clearly show the ac skin effect: at high frequency (5 kHz), the current is the highest at the edges of the conducting strip, while at a much lower frequency (250 Hz), the current density is essentially uniform. We use two vertical axes in Fig. 5 to show the surface magnetic field $ B x ( x ) $ (left) and the current density J(x) (right) that we convert from the pickup coil signal based on our calibration procedure (see Sec. III B and Appendix C). We have also plotted the theoretical expectations for the current density (solid) and surface magnetic field (dashed) and find good agreement with the data in the center portion of the strip. For completeness, we also present all of our measured data in Fig. 6. At the edges of the strip, the data is lower than the theoretical expectation, possibly due to high field curvature or misalignment of the coil in a region with a significant B_{y} component.
The amplitude of the current density and associated surface magnetic field follow a roughly exponential fall off (with an offset) from the edges towards the middle of the strip. We define the characteristic decay constant λ as the distance from the maximum magnitude position to the position where the magnitude falls to $ 1 / e $ above the minimum value at the center. Table I shows λ for the data at all the frequencies shown in Fig. 6. The theoretical values for the current density and surface magnetic field, λ_{J} and $ \lambda B th $, are extracted from numerical simulations (Silvester method^{15} for Table I). For interpolation of the data, both a fourth order polynomial and an exponential function yield the same $ 1 / e $ values for $ \lambda B \u2009 exp $. We note that the fall off constant λ is significantly larger than the skin depth δ, as expected for the lateral skin effect regime. For example, across the few kHz region of our data, we find that $ \lambda \u223c 5 \delta $. Generally, λ depends on the geometry of the strip, which we parameterize by $ W T / \delta 2 $.^{26}
Freq (Hz) .  δ (mm) .  $ W T / \delta 2 $ .  $ \lambda J / \delta $ .  $ \lambda B th / \delta $ .  $ \lambda B \u2009 exp \u2009 / \delta $ . 

250  6.37  0.31  2.3  1.8  (No fit) 
500  4.50  0.62  3.2  3.3  2.9 (1.3) 
1000  3.18  1.25  4.4  4.8  4.8 (0.3) 
2000  2.25  2.49  5.2  6.0  5.6 (0.5) 
3000  1.84  3.74  5.1  6.3  5.5 (0.9) 
4000  1.59  4.98  4.8  6.3  5.8 (1.0) 
5000  1.42  6.23  4.6  6.3  7.0 (1.1) 
Freq (Hz) .  δ (mm) .  $ W T / \delta 2 $ .  $ \lambda J / \delta $ .  $ \lambda B th / \delta $ .  $ \lambda B \u2009 exp \u2009 / \delta $ . 

250  6.37  0.31  2.3  1.8  (No fit) 
500  4.50  0.62  3.2  3.3  2.9 (1.3) 
1000  3.18  1.25  4.4  4.8  4.8 (0.3) 
2000  2.25  2.49  5.2  6.0  5.6 (0.5) 
3000  1.84  3.74  5.1  6.3  5.5 (0.9) 
4000  1.59  4.98  4.8  6.3  5.8 (1.0) 
5000  1.42  6.23  4.6  6.3  7.0 (1.1) 
In Fig. 7, we plot the phase of the pickup coil signal (relative to the x = 0 phase) vs position x across the strip for a 5 kHz current. The data clearly show that the phase of the current density varies by more than 30° across the strip, in tandem with the magnitude. In other words, for short portions of the ac cycle, the current in the center goes in the opposite direction to the current on the edges of the strip. The theory curve for the phase agrees reasonably well with the data over the breadth of the strip. Past the strip edges, the overall pickup coil signal is weaker, and the data deviate from theory, possibly due to interference in the pickup coil from other parts of the apparatus. Furthermore, in contrast with the magnitude, the phase across the conductor, when plotted, displays a modest “bump” at the center of the strip. This nonmonotonic behavior means that for brief moments in the cycle, the current at the center and along the edges of the strip goes in the same direction, but the current between these regions goes in the opposite direction. This counterflow behavior is examined in detail in Appendix B.
V. CONCLUSION
We have directly observed the ac skin effect at kHz frequencies in a rectangular aluminum strip. We have shown experimentally that the current increasingly hugs the edges of the strip as the frequency increases and that the phase of the current density varies significantly across the strip. We have calculated the theoretical distribution of the current across the strip by four different methods and find good agreement between these and the data, with modest deviations at the edge.
Conveniently, our simple experimental setup is well suited to an inclass demonstration. The setup requires standard laboratory equipment (analog controlled current source, function generator, opampbased amplifier, current sensor, and oscilloscope) and a labbuilt elongated pickup coil. A possible upgrade to the pickup coil is to use two perpendicular elongated coils so that B_{x} and B_{y} can be measured simultaneously. Such a pickup coil would provide more information when probing the edges of the strip and the circular polarization in the case of phased currents in multiple strips.
This kHzlevel work is a stepping stone towards accurate engineering of GHzlevel microwave magnetic nearfields with much smaller conducting strips (∼100 μm) on an atom chip. Based on the principle of similitude, the agreement between theory and experiment in this kHz work provides confidence that our numerical computation methods for the ac skin effect and related magnetic near field can be extended to microwave frequencies.
ACKNOWLEDGMENTS
This work was supported in part by AFOSR, NSF, DTRA, and William & Mary.
APPENDIX A: ANALYTIC FORMS
In Fig. 8, we plot C_{n} terms in the sum of Eq. (A1) for different values of $  k  $ with N = 500. Each segmented line connects consecutive C_{n} terms in sequence, beginning in the lower right with the C_{0} or dc value and ending for converged sequences with many values near the origin, contributing very little. Solutions require many powers of s for convergence at high frequency, while only a few are needed for low frequency convergence. At low frequencies (e.g., $  k  = 0.02 $), similar to the discussion in Sec. II A, the primary contribution is largely real, with a small imaginary s^{2} contribution. The leading C_{0} terms describe a semicircle of diameter $\u22480.36\u22481\u22122/\pi $, a feature also described in Casimir and Ubbink's analysis.^{8}
This function appears, scaled to our parameters, in Fig. 1. This expression also shows that in the center of a very thin strip, the minimum J will only drop to $ 2 / \pi \u2248 0.63662 $ times the dc value. Our other simulations reinforce the trend toward this curve at higher frequencies. The phase in the high frequency limit approaches a uniform distribution, lagging the driving current by 90° (for a very thin ribbon).
APPENDIX B: THE COUNTERFLOW EFFECT
Due to the phaseshifting of the skin effect, the dominant outer current always precedes the rest of the current (see Fig. 9). However, above some frequency, the phase in the center slightly precedes the area surrounding it. As seen in Fig. 7, a slight bump in phase is present at the center of the strip.
As shown in Fig. 9 (3, 4, and 5 kHz curves), for a small portion of time (about 1°, twice per cycle), the currents in the center and edge go in the same direction, but currents at points between flow in the opposite direction, near the zero crossing of a current oscillation. Our calculations show that this effect begins at $  k  \u2248 2.0514 $, which corresponds to about 2.6 kHz in our experiment. The effect grows and then diminishes at very high frequencies, and we predict no higher order phase reversals.

If $ a \u2033 /a> b \u2033 /b$, the center phase always follows locally.

If $ a \u2033 /a= b \u2033 /b$, the center is in phase with its surroundings.

If $ a \u2033 /a< b \u2033 /b$, the center phase precedes locally.
Using the summation $ J ( s ) = C 0 + C 1 s 2 + C 4 s 4 + \cdots $, we can identify $ C 0 = a + i b  s = 0 $ and $ C 1 =1/2!(a\u2033+ib\u2033)  s = 0 $. We only need to compute two complex terms of Eq. (A1)'s solution, C_{0} and C_{1}, to know whether the current has this phase reversal at the center. The equality condition $ a \u2033 / a = b \u2033 / b $ implies that C_{0} and C_{1} lie on a line through the origin (see the dashed line in Fig. 8). Numerically, we find that this happens for $  k  = 2.0514 \u2248 2 $. The frequency required for this central phase reversal effect is then roughly $ f \varphi \u22731/WT\mu \sigma $. At higher frequencies, the effect is at most only a few degrees, which is sufficient for observation (see Fig. 7).
APPENDIX C: CALIBRATION
In principle, we can extract values of J(x) or B(x) via Eq. (5), a measurement of $ V coil ( x ) $, the frequency ω, and the coil area A, but this approach is problematic. First, the coil area is not well defined because the coil's enclosed area is comparable to the wire area (see Figs. 3(b) and 3(c)). Second, the signal we measure on the oscilloscope V_{sig} is also modified by the bandwidth of the amplifier. Finally, V_{coil} may have additional magnetic gradient dependence or ωdependence beyond the linear ω scaling in Eq. (3).
The proportionality constants $ \alpha J ( \omega ) $ and $ \alpha B ( \omega ) $ are determined via calibration experiments at known J and B_{surf}. These two constants also have the same frequency dependence, so for two different frequencies ω and ω_{0}, we expect $ \alpha J ( \omega ) / \alpha J ( \omega 0 ) = \alpha B ( \omega ) / \alpha B ( \omega 0 ) $. From this relation, we see that we have $ \alpha J ( \omega ) = \alpha J ( \omega 0 ) ( \alpha B ( \omega ) / \alpha B ( \omega 0 ) ) $, so we can obtain $ \alpha J ( \omega ) $ from measurements of $ \alpha J ( \omega 0 ) $ and $ \alpha B ( \omega ) / \alpha B ( \omega 0 ) $.
We measure $ \alpha J ( \omega 0 ) $ at a very low frequency with $ \omega 0 = 2 \pi \xd7 \u2009 250 $ Hz, where the ac skin effect is near negligible yet high enough in frequency to be visibly picked up by the coil. The current density J(x) is nearconstant across the middle of the strip as seen in Fig. 5.
We determine $ \alpha B ( \omega ) / \alpha B ( \omega 0 ) $, i.e., the frequency dependent gain of the pickup coil system, by measuring V_{sig} at the surface of an aluminum rod of the circular cross section driven by a known ac for different frequencies. Due to its geometry, the external magnetic field of the rod is frequency independent (unlike the strip), so we can use $ \alpha B ( \omega ) / \alpha B ( \omega 0 ) = ( V sig ( \omega 0 ) / \omega 0 ) / ( V sig ( \omega ) / \omega ) $ to determine the $ \alpha B ( \omega ) / \alpha B ( \omega 0 ) $ calibration ratio.
REFERENCES
This calculation was done using formulas from the paper of Smith.^{29} The formulas are the following (they are approximations for the high frequency limit, i.e., wire radius $\u226b$ skin depth): $Rac/Rdc=(1/2)\u2009(r/\delta ),\u2004(Lac/Ldc)=1/(0.5\u2009*\u2009(r/\delta ))$, with $\delta =2/\sigma \mu \omega $.
We learned of this work after submission of the manuscript.
The infinite halfspace conductor can also be solved analytically.^{32}
Both FEKO and Flux are distributed by Altair Inc.
Older versions of FEKO (before 2019) appeared to use a pseudorandom arrangement of mesh elements, which led to overall continuous and comparable results. Newer versions use regions of aligned rows of uniform triangle mesh elements, and we observe deviations, such as those evident in the amplitude and phase at the center of the strip in Fig. 1, occurring at the borders between these uniform regions.