The field near a point, line, or surface charge example 16
.Compare the variation of the electric field with distance,d, for
small values ofdin the case of a point charge, an infinite line of
charge, and an infinite charged surface.
.For a point charge, we have already foundE ∝ d−^2 for the
magnitude of the field, where we are now usingdfor the quantity
we would ordinarily notate asr. This is true for all values ofd,
not just for smalld— it has to be that way, because the point
charge has no size, so ifEbehaved differently for small and large
d, there would be no way to decide whatd had to be small or
large relative to.
For a line of charge, the result of example 13 is
E=
kλL
d^2√
1 +L^2 / 4 d^2.
In the limit ofd L, the quantity inside the square root is domi-
nated by the second term, and we haveE∝d−^1.
Finally, in the case of a charged surface, the result is simplyE=
2 πσk, orE∝d^0.
Notice the lovely simplicity of the pattern, as shown in figure j. A
point is zero-dimensional: it has no length, width, or breadth. A
line is one-dimensional, and a surface is two-dimensional. As the
dimensionality of the charged object changes from 0 to 1, and
then to 2, the exponent in the near-field expression goes from 2
to 1 to 0.
Section 10.3 Fields by superposition 603