Page 32 - Engineering Electromagnetics, 8th Edition

14 ENGINEERING ELECTROMAGNETICS

The circular cylindrical coordinate system is the three-dimensional version of
the polar coordinates of analytic geometry. In polar coordinates, a point is located
in a plane by giving both its distance ρ from the origin and the angle φ between the
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line from the point to the origin and an arbitrary radial line, taken as φ = 0. In
circular cylindrical coordinates, we also specify the distance z of the point from an
arbitrary z = 0 reference plane that is perpendicular to the line ρ = 0. For simplicity,
we usually refer to circular cylindrical coordinates simply as cylindrical coordinates.
This will not cause any confusion in reading this book, but it is only fair to point out
that there are such systems as elliptic cylindrical coordinates, hyperbolic cylindrical
coordinates, parabolic cylindrical coordinates, and others.
We no longer set up three axes as with rectangular coordinates, but we must
instead consider any point as the intersection of three mutually perpendicular sur-
faces. These surfaces are a circular cylinder (ρ = constant), a plane (φ = constant),
and another plane (z = constant). This corresponds to the location of a point in a
rectangular coordinate system by the intersection of three planes (x = constant, y =
constant, and z = constant). The three surfaces of circular cylindrical coordinates are
shown in Figure 1.6a. Note that three such surfaces may be passed through any point,
unless it lies on the z axis, in which case one plane suffices.
Three unit vectors must also be defined, but we may no longer direct them along
the “coordinate axes,” for such axes exist only in rectangular coordinates. Instead, we
take a broader view of the unit vectors in rectangular coordinates and realize that they
are directed toward increasing coordinate values and are perpendicular to the surface
on which that coordinate value is constant (i.e., the unit vector a x is normal to the
plane x = constant and points toward larger values of x). In a corresponding way we
may now define three unit vectors in cylindrical coordinates, a ρ , a φ , and a z .
The unit vector a ρ at a point P(ρ 1 ,φ 1 , z 1 )is directed radially outward, normal
to the cylindrical surface ρ = ρ 1 .It lies in the planes φ = φ 1 and z = z 1 . The unit
vector a φ is normal to the plane φ = φ 1 , points in the direction of increasing φ, lies in
the plane z = z 1 , and is tangent to the cylindrical surface ρ = ρ 1 . The unit vector a z
is the same as the unit vector a z of the rectangular coordinate system. Figure 1.6b
shows the three vectors in cylindrical coordinates.
In rectangular coordinates, the unit vectors are not functions of the coordinates.
Twoof the unit vectors in cylindrical coordinates, a ρ and a φ ,however, do vary with
the coordinate φ,as their directions change. In integration or differentiation with
respect to φ, then, a ρ and a φ must not be treated as constants.
The unit vectors are again mutually perpendicular, for each is normal to one of the
three mutually perpendicular surfaces, and we may define a right-handed cylindrical

4 The two variables of polar coordinates are commonly called r and θ.With three coordinates,
however, it is more common to use ρ for the radius variable of cylindrical coordinates and r for the
(different) radius variable of spherical coordinates. Also, the angle variable of cylindrical coordinates is
customarily called φ because everyone uses θ for a different angle in spherical coordinates. The angle
φ is common to both cylindrical and spherical coordinates. See?