Simple Nature - Light and Matter

10.2.2 Two or three dimensionsThe topographical map in figure a suggests a good way to visualizethe relationship between field and voltage in two dimensions.Each contour on the map is a line of constant height; some of theseare labeled with their elevations in units of feet. Height is relatedto gravitational energy, so in a gravitational analogy, we can thinkof height as representing voltage. Where the contour lines are farapart, as in the town, the slope is gentle. Lines close together indicatea steep slope.If we walk along a straight line, say straight east from the town,then height (voltage) is a function of the east-west coordinate x.Using the usual mathematical definition of the slope, and writingV for the height in order to remind us of the electrical analogy, theslope along such a line is dV/ dx (the rise over the run).What if everything isn’t confined to a straight line? Water flowsdownhill. Notice how the streams on the map cut perpendicularlythrough the lines of constant height.It is possible to map voltages in the same way, as shown infigure b. The electric field is strongest where the constant-voltagecurves are closest together, and the electric field vectors always pointperpendicular to the constant-voltage curves.The one-dimensional relationship E = − dV/ dx generalizes tothree dimensions as follows:E x = − dVdxE y = − dVdyE z = − dVdzThis can be notated as a gradient (page 215),E = ∇V ,and if we know the field and want to find the voltage, we can use aline integral,∫∆V = E · dr ,where the quantity inside the integral is a vector dot product.Cself-check CImagine that figure a represents voltage rather than height. (a) Considerthe stream the starts near the center of the map. Determine thepositive and negative signs of dV / dx and dV / dy, and relate these tothe direction of the force that is pushing the current forward against theresistance of friction. (b) If you wanted to find a lot of electric charge onthis map, where would you look? ⊲ Answer, p. 925a / A topographical map ofShelburne Falls, Mass. (USGS)b / The constant-voltage curvessurrounding a point charge. Nearthe charge, the curves are soclosely spaced that they blendtogether on this drawing due tothe finite width with which theywere drawn. Some electric fieldsare shown as arrows.Section 10.2 Voltage Related To Field 571