A new technique is presented for understanding how charges distribute themselves on the surface of a conductor during current flow. The technique uses a set of three-dimensional calculation cells (“pixels”) that cover the conductor's surface and contain internal charge. The pixels have two faces separated by an infinitesimal, but finite, distance, with one face being conductive and the other non-conductive. Each pixel acts as a sensor by responding to (“sensing”) the net Coulomb electric field at its conductive face due to charges in other pixels and charges at the current source and sink (collectively, the external charges). Through a feedback process implemented as a series of time steps, the pixels' internal charges adjust themselves until, at each pixel, a balance is achieved between the electric flux at the conductive face due to the external charges and that due to the pixel's internal charge. Specifically, at each time step, for each pixel at which there is flux imbalance, charge will move into or out of the pixel's conductive face in the direction that reduces the imbalance. The charge distribution for the set of pixels that gives balanced flux for each of the pixels is the system's steady state and for systems where retardation effects are not significant, e.g., biological systems, the time series is the path by which the system reaches that state. Fluxes are calculated using solid angles and because solid angles do not vary with a change in scale, the charge distribution, when expressed in terms of charges/pixel, as opposed to charges/area, depends only on the system shape.

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20.
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23.
An analogy to optical sensors may be helpful to some students in answering the question “what would a sensing pixel sense.” Both optical sensors and sensing pixels sense close sources better than distant sources and head-on sources better than oblique sources. Quantitatively, the response of an optical sensor to a light source depends on the solid angle of the sensor as seen from the source. See, for example,
Thilo
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Which way does the light go?
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Am. J. Phys.
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24.
Jonas
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25.

See Ref. 12, p. 1188. As discussed therein, retardation effects can be ignored when τsysτc, where τc is the time it takes light to cross the system and τsys is the system's relaxation time, which depends on both the system's geometry and the relaxation time (τmed) of the system's conductive medium. As illustrated in Ref. 12, τsys can be hundreds of times greater than τmed thus allowing retardation effects to be ignored even though locally the system relaxes very fast based on the value of τmed. If retardation effects cannot be ignored, then in the equations of Sec. II A, q(tr/c) would need to be used instead of q(t).

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28.
That a charge in the same plane as a sensing pixel has essentially no effect on the sensing pixel can be seen from: (1) the r ̂ · n ̂ dot product in the equation in Table I which approaches zero for both the conductive and non-conductive faces of the pixel as the pixel's thickness approaches zero since the faces become co-planar with r ̂ and thus their normals n ̂ become perpendicular to r ̂ , causing the dot product to become zero; and (2) the solid angles of the pixel's rectangular side faces which approach zero as the thickness of the pixel approaches zero as can be seen from the equation for the solid angle of a rectangle given in, for example,
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29.

In Fig. 4, the conductive and non-conductive faces of the pixels are perpendicular to the plane of the figure. The thicknesses of the pixels have been exaggerated for purposes of illustration. For each pixel, the pixel's conductive face lies just inside the conductor and the pixel's non-conductive face is part of the bounding surface of the surrounding non-conductor. As discussed in connection with the derivation of Eq. (14), the actual pixel thickness is sufficiently small so that the solid angles of the sides of the pixel are insignificant relative to the solid angle of the pixel's conductive and non-conductive faces.

30.

We refer to a change in the charge in one pixel (the source pixel) “seeking to change” the charge in another pixel (the target pixel), rather than simply saying that the change in the source pixel “changes” the charge in the target pixel because through the interplay of all of the pixels of the system with the target pixel, as well as the effects of the charges at the source and sink, the charge in the target pixel is not merely that which the source pixel would produce in the target pixel if the source pixel were acting alone. Rather, the charge in the target pixel depends on the net effect at the pixel's conductive face of the Coulomb fields of all of the external charges. Thus, while a change in charge in the source pixel seeks to change the charge in the target pixel in a particular direction, it does not necessarily achieve that result because the effects of other charges in the system may be stronger and cause the target pixel to end up changing its charge in the opposite direction.

31.
An analytic expression for the solid angle of a square is given in
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32.
The elevated edge charges and corner flares of Fig. 5 are examples of the singularities in charge levels that occur at the intersection of conducting planes. See
J. D.
Jackson
,
Classical Electrodynamics
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Wiley
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New York
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), p.
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; see also Ref. 10, p. 862. As shown in Fig. 7, when functioning as receivers of electric flux, the pixels extend to the edges of the cube. However, when functioning as sources of electric flux, the pixel charges are assumed to be located at the centers of the pixels. This, in effect, rounds the cube's corners, the rounding being around 0.02 times the cube's edge length for the pixelization of Fig. 7. With finer meshes, the edges and corners of the cube become sharper, thus producing larger edge charges and corner flares. Amplified edge charges are shown and discussed in Jackson (Ref. 10) in connection with Jackson's analysis of his central conductor with a gap. Amplified edge and corner charges can also be clearly seen in the figures of Müller (Ref. 13), especially near the poles of Müller's battery, and in Figs. 4 and 17 of Chabay and Sherwood (Ref. 14).
33.

The potentials shown in Fig. 8 were calculated from charge distributions calculated using Eqs. (11)–(12) and (16) and the technique of Sec. 4.2.4 of Makarov et al. (Ref. 3) to convert the charge distributions into potential distributions. 1,200 triangular pixels were used for the cube and 1,280 for the sphere; Δt was set equal to 0.1τmed and 250 time steps were used to reach steady state. For clarity, the equipotential lines outside the cube and sphere are not plotted in Fig. 8. Those lines are continuous with the internal lines at the conductor/non-conductor boundaries with “kinks” in the lines at the boundaries for the plots of the second and third columns of the figure due to the presence of the surface charge as discussed in Müller (Ref. 13).

34.
See, for example,
J. H.
Jeans
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The Mathematical Theory of Electricity and Magnetism
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,
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,
1925
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(Fig. 78);
J. A.
Stratton
,
Electromagnetic Theory
(
McGraw-Hill
,
New York
,
1941
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205
207
(Fig. 37b);
M.
Zahn
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(
Wiley
,
Hoboken, New Jersey
,
1979
), p.
291
(Fig. 4–12(a)); and Ref. 32, pp. 157–159.
35.
The magnitudes of all the charges can, of course, be changed by changing the magnitudes of the sources, i.e., the Iappl value in
Eqs. (11), (12), and (20). For systems where retardation effects are significant, the steady state charge distribution will be invariant to a change in scale, but the time needed to reach steady state will be extended as the system is made larger.
36.
B.
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Drawings of pyramidal cells from the late 1800's and early 1900's by the Nobel-prize winning physiologist Ramòn y Cajal can be found
in
The Beautiful Brain: The Drawings of Santiago Ramòn y Cajal
, edited by
Eric A.
Newman
,
Alfonso
Araque
, and
Janet M.
Dubinsky
(
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,
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37.
M.
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38.
E.
Frank
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Electric potential produced by two point current sources in a homogeneous conducting sphere
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Related analytic expressions, less convenient for the derivation of Eq. (A6), can be found in
W. R.
Smythe
,
Static and Dynamic Electricity
, 3rd ed. (
McGraw-Hill
,
New York
,
1968
), pp.
259
260
.
39.
See, for example, Ref. 32, p. 101.
40.
Reference 16, p. 89.
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