Some potential fields can be made visible by simple procedures. A familiar example is the use of iron filings to show the magnetic field of a horseshoe magnet. The magnet is placed on a level, nonmagnetic surface and covered with a sheet of paper. When the filings are sprinkled on the paper, they form a pattern of curving lines that trace the magnetic field in two dimensions. Some potential fields can be investigated mathematically and by graphical means. Others can be simulated optically with moiré patterns.

Perhaps the most versatile and inexpensive instrument available to amateurs for depicting and analyzing potential fields of many kinds is the fluid mapper. The device simulates potential fields by thin lines of dye that form in the flow of a sheet of water or some other fluid representing the phenomenon under investigation. In addition to the lines of dye, portions of the fluid can be colored to identify selected areas in the field. The color not only increases the power of the fluid mapper as an analytical tool but also transforms it into an engrossing plaything with which one can generate abstract designs that resemble the colorful abstractions made by harmonographs and kaleidoscopes [left].

The fluid-flow method of simulating years ago by H. S. Hele-Shaw of the University of Oxford. His ideas suffered a long period of neglect. A. D. Moore, now professor emeritus of electrical engineering at the University of Michigan, revived the method some years ago; in the past two decades he has devised numerous techniques for making and operating fluid mappers. Moore explains the construction and use of the devices as follows:

"The fluid mapper consists essentially of a pair of closely spaced plane surfaces immersed horizontally in a shallow pan of water. The upper surface might consist of a sheet of plate glass and the lower one of a specially shaped slab of plaster. Usually the surfaces are spaced about .035 inch apart, but flow spaces of half or twice that can be used. The plaster slab may have in it one or more holes of various shapes and sizes. Some of them may be partly filled with sand, depending on the nature of the potential field that is to be depicted.

"The holes, known as wells, are usually connected at the bottom to rubber tubes for admitting a controlled flow of water from movable reservoirs, which ordinarily are quart-sized cans equipped at the bottom with nipples that fit the rubber tubes. Flow is generated in the space between the plaster slab and the glass plate by moving the reservoirs above or below the level of the water in the pan. Water within the thin flow space moves either in the direction of the forces that constitute the potential field being simulated or at right angles to the forces, depending on the nature of the field. The direction of flow within the thin space is made visible by inserting fixed particles of dye between the glass and the plaster slab when the apparatus is assembled. Typically the dye consists of small crystals of potassium permanganate, which dissolve slowly to form thin purple trails in the water.

"The flow space between the plaster slab and the glass functions as a scale model of the potential field under investigation. The field of a horseshoe magnet, for example, can be simulated by a rectangular flow space with two wells that represent the poles of the magnet. The reservoirs that supply the wells can be manipulated so that one well functions as a source of flowing water and the other as a sink. Dye lines in the flow space surrounding the wells portray the potential field, in this case the magnetic flux.

"Experimenters sometimes need to know the electrical resistance of an irregularly shaped sheet of conducting material. An example might be the resistance from the center to the edges of a sheet of copper in the shape of a keystone. The problem can be solved easily by means of a fluid mapper, which shows the path the current would take through the copper. The example will illustrate the procedure of constructing and operating a fluid mapper.

"A well in a keystone-shaped slab of plaster represents the electrical connection to the inner edge of the copper sheet [see Figure 2]. The outer boundary of the slab represents the outer edge connection. The slab is equipped with four plaster feet, which rest on a horizontal sheet of plate glass. A second horizontal sheet of plate glass is supported .035 inch above the plaster slab by three spacers, which also rest on the lower sheet of plate glass. The lower end of the well is closed except for a rubber tube that admits water to the well from a reservoir in the form of a tin can.

"The plaster slab is made first. Almost any plaster can be used, but after much experimentation I found that the best material is the plaster commonly used by dentists in making dentures. I prefer the type known as Duroc, a product of the Ransom and Randolph Company of Toledo, Ohio. Duroc comes as a yellow powder; when it is mixed to the consistency of soft paste, it sets to stone hardness in from 10 to 20 minutes. The cost varies roughly from 15 to 25 cents per pound (or pint) of powder, depending on the quantity bought.

"Typically the slabs are cast in shallow, boxlike molds that have bottoms of plate glass and sides that can consist of narrow strips of metal temporarily fixed to the glass on the outside by dabs of plaster. Slabs with curving sides, such as the one shown in the accompanying illustration [above left] can be formed of flexible strips. Usually I solder a short, right-angled strip to one side of each straight rail near the center; the strips serve both as handles for manipulating the rails and as braces.

"To make a slab in the form of a keystone I cut a template of cardboard to the planned size, say eight inches long and six inches wide at the broad end [see Figure 4]. A square hole is cut in the cardboard in order to admit a cylindrical ring for molding the well. Such inserts are usually made with a slight taper and greased with Vaseline for easy removal from the hardened plaster. I center the template on a level sheet of plate glass, place the cylindrical ring in the square hole of the template (holding the ring with the narrow end down) and cement the ring in place by pouring in some plaster. The metal rails are placed snugly against the four sides of the template and similarly cemented in place. After the plaster has set I remove the template and pour the slab. Slabs approximately 3/8 inch thick work nicely. Larger slabs are made thicker.

"The quantity of plaster required for casting a slab is determined by the volume of the slab. For the slab I have been describing the volume is about 12 cubic inches. A pint of Duroc powder, when added to six ounces of water, yields plaster of about optimum consistency. Pour the powder into the water and stir the mixture with a spoon until the lumps are gone.

"Ladle about two-thirds of the mix into the mold and spread it around with a spoon. Air bubbles will be trapped in the corners and on the upper surface of the plate glass. They must be dislodged or indentations will appear in the finished slab. To remove the bubbles probe the wet mix with a small stiff brush. Jiggle the brush up and down, particularly in the corners and along the rails. The air bubbles will rise into the mix, where they will do no harm.

"Add and distribute the rest of the plaster. Agitate the entire surface with the bowl of a spoon, using a jiggling up-and-down motion. As a result of this procedure the mix flows into a relatively smooth upper surface. After about 20 minutes the slab will have hardened adequately.

"Wet the whole assembly in a container of water or under a tap. Absorbed water will release the plaster from the glass. The plaster will slide off the glass readily. The cylindrical ring can now be removed. Invert the slab. The face that was in contact with the glass should appear glossy.

"The slab is now equipped with supporting legs. This operation requires the use of two sheets of plate glass and three cylindrical spacers about an inch long. The length of the spacers is not critical, but all three must be equal in length to within .0005 inch. They can be made of metal or plaster.

"Select three wood screws that have flat heads and, with a file, dress down the points of the two longer screws until they match the shortest one as closely as possible. Make three cylinders of gummed paper tape about 3/4 inch wide and 3/4 inch long. Center the paper cylinders upright on a level sheet of plate glass. Stand the wood screws on their heads near the edges of the glass in the largest possible triangular configuration. Place the cylinders close together at the center. Fill them with plaster, heaping the mix on top to a height of about 1k inches. Lower the second glass plate horizontally over the first until it rests on the three wood screws. After 20 minutes wet the castings to release them from the glass. These spacers may be enough alike to be used. If not, repeat the procedure, except this time substitute the plaster spacers just made for the three wood screws. Discard the spacers that were made first. The second set will be almost identical in length.

"Wash and dry the two glass plates. Place one of the plates on a level surface. Center the keystone slab on the plate with the glossy side down. Place the plaster spacers in a triangular array near the edges of the glass. Thoroughly wet the slab but mop off standing water. Apply small dabs of plaster to the four corner areas. These dabs will become a set of feet. Add more plaster on top of each to make a pile about 1 1/4 inches high.

"Lower the second sheet of plate glass horizontally over the first until it comes to rest on the spacers. The glass will press the piled plaster to a height that matches the height of the spacers. When the plaster has set firmly, wet it and separate the plate from the feet.

"The slab is now equipped with its own permanent set of four feet. The well must next be closed at the bottom and fitted with inlet tubing. Place the slab, glossy side down, on a glass plate. Cut a four-inch length from a piece of rubber tubing with an inside diameter of approximately 1/8 inch. I use the type of tubing called handmade tubing, which is characterized by a rough outside surface.

"From thin sheet aluminum, heavy tinfoil or wet cardboard cut a disk about half an inch larger in diameter than the well in the slab. Heavily coat one end of the rubber tubing with plaster for half an inch or so (being careful not to plug it) and lay it on the slab with the plastered end at the edge of the well. Bend up the disk or patch at one edge to make it fit roughly down over the tubing, place it over the well and apply plaster all over it. Be sure to seal it all around to prevent leakage. This operation completes the slab.

"A sheet of plate glass is centered in the bottom of a rectangular baking pan, preferably one made of aluminum or some other nonrusting metal. Place the completed slab, glossy side up, on the plate glass. Arrange the spacers around the slab in triangular array and fairly close to the edges of the plate glass. The tops of the spacers are now flush with the glossy surface of the slab. They will support a second sheet of plate glass above the slab, but the length of the spacers must be increased to provide a flow space about .035 inch thick between the slab and its glass cover. This is done by placing a shim on top of each spacer. I use brass washers 7/16 inch in diameter for shims and adjust their thickness by rubbing them on grit paper.

"I start with seven washers. Both sides of all seven are first ground lightly to remove burrs. Three are then placed in a widely spaced triangular array between the two sheets of plate glass. The remaining four are slid one at a time between the plates to test for fit. The three thinnest washers are substituted as the plate separators. Thereafter, by cut and try, the remaining washers are alternately ground and tested. Ultimately the three that are most nearly equal in thickness are selected for use.

"After completing the shims I solder a nipple into the bottom of a quart-sized tin can. The nipple is connected to the well inlet by a 30-inch length of rubber tubing of the same size and type as the tubing used for the well inlet. For the coupling I use a short length of metal tubing

"Pour enough water into the pan to immerse the apparatus to a depth of about half an inch. Put an inch or two of water in the can. Alternately raise and lower the can to force out any bubbles trapped in the tubing. Sometimes one must use a rubber ear syringe to force water in and out of the tubing and thus dislodge firmly trapped air. Some water contains a large amount of absorbed air that must be removed by boiling or prolonged standing.

"Dye in the form of crystals of potassium permanganate is now prepared. If the crystals appear to be larger than about one millimeter, grind them lightly in a mortar. Sift the grindings through two screens, one with spacings slightly less than one millimeter in diameter and the other with spacings slightly larger. A tea strainer can serve as the finer screen. Crystals that remain in the finer sieve but pass through the coarser sieve are transferred to a saltshaker in which all holes but one are covered by adhesive tape. Warning: Potassium permanganate is toxic. Do not inhale the dust or get it in the eyes.

"Sprinkle the water above the immersed slab with a circle of crystals spaced about five millimeters apart. A circle that lies midway between the well and the edge of the slab will yield a nice field pattern. The crystals sink through the water and come to rest on the slab. Try various sprinklings for the best effects on a new slab.

"Finally, lower a sheet of plate glass gently on the shimmed spacers. Avoid abrupt movements that would disturb the water enough to wash the crystals off the slab. Incline the glass slightly so that one edge enters the water first; in this way the formation of bubbles on the lower surface of the glass is minimized. When this cover glass is in place, slowly lower the reservoir a few inches. Thin lines of dye will flow from the crystals to the well. Return the reservoir to its former position and then slowly elevate it somewhat. The lines of dye will flow from the well to the crystals. In addition, lines will now flow from the crystals to the edges of the slab. Thus by manipulating the reservoir one can cause a complete pattern of the potential field to extend from the well to the edges of the keystone.

"The pattern can be photographed. Alternatively, the lines of dye can be made to register permanently on the slab itself. I discovered this technique quite by accident. (Indeed, many of the other procedures were learned by trial and error.) On one occasion I applied a thin coat of rubber-emulsion paint to a slab, hoping to prevent the erosion of the glossy face. After operating the apparatus with potassium permanganate dye I was surprised and pleased to observe that the dye lines reacted chemically with the paint and made a permanent imprint. At present I use Sherwin-Williams' Super Kem-Tone (white) for-this kind of record-making. A record of the pattern made by a keystone slab is depicted in the accompanying illustration [bottom left].

"With this pattern at hand it is relatively easy to determine the electrical resistance that would be introduced in a circuit by a conductor of this shape. To determine the numerical data from the pattern I place a sheet of translucent paper over the slab and trace about 24 lines that happen to form a radial pattern of reasonably uniform spacing. If the resulting figure, which I call a 'map' of the field, is characterized by symmetry, as in the case of a keystone, only half of the pattern need be traced [see illustration at bottom right].

"The space between adjacent flow lines of the map is then subdivided into increments known as curvilinear squares. Two

properties characterize the curvilinear square. First, approximately but usually very closely, each side will touch an inscribed circle at some point. Second, as a curvilinear square is successively subdivided into smaller and smaller increments-into quarters, eighths, sixteenths and so on-progressively smaller increments approach more and more closely the shape of a true square.

"This property is particularly useful because true squares of any size cut from a given sheet of conducting material are equal in electrical resistance. Assume, for example, that opposite edges of a conducting square are connected in an electrical circuit and that one volt applied across the square will induce in the square a current of one ampere. The resistance of the square is equal to the quotient obtained by dividing the voltage by the current; in this case the resistance is one ohm.

"Assume that the sheet is divided down the middle so that half of the current is confined to each half of the sheet. A measurement of the voltage would show half of the voltage across each half of the resulting rectangular strip. Each quarter of the original square would then have imposed across it .5 volt that would induce a current of .5 ampere. The resistance of each quarter of the square would equal one ohm, because 1/2 divided by 1/2 is 1. Because curvilinear squares approach true squares in shape when they are sufficiently subdivided, the electrical resistance of all curvilinear squares is independent of their size if the squares are all cut from the same material.

"Three pairs of flow lines form three tubes; these have been mapped into curvilinear squares in the illustration below. I made the subdivisions with the aid of a drawing instrument known as a circle template. It is a sheet of plastic with a series of holes of graduated diameter. Such templates can be bought in stores that deal in artists' and engineers' supplies.

"The template is placed over a tube. Through an opening of appropriate size I draw a circle that is tangent to the flow lines and also tangent to the end, after which the resulting curvilinear square is closed by drawing a line that is also tangent to the circle. The template is then shifted for similarly subdividing the rest of the area. Subdivision can proceed from either end or both ends. Occasionally a tube will divide exactly into a whole number of curvilinear squares, as at A in the illustration. More often a fractional rectangular area will remain. The remainder is subdivided just as though its sides were flow lines, as at B and C.

"Electrical resistance is evaluated by analyzing the subdivided map; other physical quantities can also be studied in this way. In the case of electrical resistance each curvilinear square is tentatively assumed to have a resistance of one ohm. I call this a 'relative' ohm. The value is later converted to true ohms. In making the evaluation curvilinear squares are treated as though they were resistors in a circuit.

"The total resistance of a number of adjacent curvilinear squares in series is equal to their sum. The total resistance of a group of curvilinear squares in parallel is equal to the reciprocal of their sum. The string of seven curvilinear squares enclosed by the flow lines at A in the illustration, for example, has a total electrical resistance of seven relative ohms. The total resistance of the tube at B, which has three full curvilinear squares and is also connected to the three little parallel curvilinear squares that constitute the fractional remainder, is equal to 3 plus 1/3, or 3 1/3 relative ohms. At C the total resistance amounts to three relative ohms plus one relative ohm in parallel with two relative ohms in series, or 3 2/3 relative ohms.

"The total relative resistance of one half of the symmetrical keystone is found by adding the reciprocals of all the relative ohms of all tubes on that side, then taking the reciprocal of that value. The total relative resistance of the entire keystone is equal to half of the resistance of one side of it because the sides are in parallel.

"To convert relative resistance into true resistance cut a true square from the conducting material of which the keystone will be made. Measure or otherwise determine the true resistance of the true square. Multiply the total relative resistance of the keystone, as determined by the fluid-mapping technique, by the actual resistance per square of the conducting material. The resulting product is equal to the true resistance of a keystone made of that material. Electrical capacitance, heat conductivity and like physical quantities can be similarly determined by substituting appropriate equations and data.

"Occasionally an experimenter wishes to analyze the potential field of a distributed source, such as when heat is developed in the whole volume of a conductor or in an electrical coil, the heat having to flow to the surface and then be conducted away through surrounding material with the same thermal conductance. Fields associated with such distributed sources are simulated by a well filled with sand or fine metallic shot. The sand or shot rests on braced screening placed partway down the well; the surface of the sand bed is made exactly flush with the upper surface of n the plaster slab. The sand would simulate the cross section of the coil. Other variations in fluid-mapper technique include adjustments in the thickness of the flow space to simulate discontinuities in conductivity.

"Slabs equipped with multiple wells are frequently used to simulate interactions of potential fields. An example is the interaction of electrostatic fields in a radio tube equipped with three electrodes: a cathode, a grid and a plate. A circular slab would represent a horizontal cross section through the vacuum tube. A central well would simulate the negatively charged cathode. The edge of the slab would represent the positively charged plate. A circle of six equal intermediate wells would represent the grid electrode.

"Water, colored green by vegetable dye, would flow from the central well. Water colored red would similarly flow from the six interconnected 'grid' wells. Dye lines formed by potassium permanganate crystals would depict the direction of the electrostatic forces within the tube. A fluid map made by a slab of this type appears on the cover of this issue. The relative potentials and polarities imposed on the simulated electrodes are varied by adjusting the relative heights of the reservoirs with respect to the level of clear water in the pan.

"Colored water sometimes comes up in the pan, goes over the plate and obscures the flow pattern. One can prevent this by using water in the reservoirs that is slightly cooler than the water in the pan.

"Having made a numerical analysis of the potential fields in the vacuum tube, the experimenter may wish to explore a lighter side of the apparatus: its potential for generating abstract designs in color. This is accomplished by randomly changing the elevation of the reservoirs with respect to the water level in the pan. The accompanying photographs [top of page] of the vacuum-tube mapper depict six patterns that appeared during a constantly changing color show that lasted 15 minutes."

Bibliography

FLUID MAPPER MANUAL. A. D. Moore. The University of Michigan, 1961.

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