Inside Collection: A First Course in Electrical and Computer Engineering

This module is part of the collection, *A First Course in Electrical and Computer Engineering*. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.

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 the components 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 distribution
of 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 all
such infinitesimal charges. This idea is illustrated in Figure 1, where the
charge

But the field strength is directed at angle

with components

For charge uniformally distributed with density

The functions cos

We leave it as a problem to show that the real component

We emphasize that the field at *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 vertical
component of the real field.

Show that the horizontal component of the field

From the symmetry of this problem, we conclude that the field around the infinitely long wire of Figure 1 is radially symmetric. So, in polar coordinates, we could say

which is independent of

An electric field has units of volts/meter, a charge density

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