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We have established that vectors may be used to code complex numbers. Conversely, complex numbers may be used to code or represent the orthogonal components of any two-dimensional vector. This makes them invaluable in electromagnetic field theory, where they are used to represent thecomponents of electric and magnetic fields.
The basic problem in electromagnetic field theory is to determine the electric or magnetic field that is generated by a static or dynamic distributionof charge. The key idea is to isolate an infinitesimal charge, determine the field set up by this charge, and then to sum the fields contributed by allsuch infinitesimal charges. This idea is illustrated in [link] , where the charge , uniformly distributed over a line segment of length at point , produces a field at the test point . The field is a “vector” field (as opposed to a “scalar” field), with components and . The intensity or field strength of the field is
But the field strength is directed at angle , as illustrated in [link] . The field is real with components and , but we code it as a complex field. We say that the “complex” field at test point is
with components and . That is,
For charge uniformally distributed with density along the x-axis, the total field at the test point is obtained by integrating :
The functions cos and are
We leave it as a problem to show that the real component of the field is zero. The imaginary component is
We emphasize that the field at is a real field. Our imaginary answer simply says that the real field is oriented in the vertical direction because we have used the imaginary part of the complex field to code the verticalcomponent of the real field.
From the symmetry of this problem, we conclude that the field around the infinitely long wire of [link] is radially symmetric. So, in polar coordinates, we could say
which is independent of . If we integrated the field along a radial line perpendicular to the wire, we would measure the voltage difference
An electric field has units of volts/meter, a charge density has units of coulombs/meter, and has units of coulombs/volt-meter; voltage has units of volts (of course).
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