ELECTROMAGNETICS
(EE 340)
CHAPTER 2
VECTOR ANALYSIS:
COORDINATE SYSTEMS
AND TRANSFORMATION
DR HOUSSEM BOUCHEKARA
INTRODUCTION
• In general, the physical quantities we shall be dealing with in EM are functions of
space and time. In order to describe the spatial variations of the quantities, we must
be able to define all points uniquely in space in a suitable manner.
• This requires using an appropriate coordinate system.
• A point or vector can be represented in any curvilinear coordinate system, which may
be orthogonal or nonorthogonal. Nonorthogonal systems are hard to work with and
they are of little or no practical use.
• An orthogonal system is one in which the coordinates arc mutually
perpendicular.
INTRODUCTION
• In three dimensions, the coordinate system can be specified by the intersection of
three surfaces.
• In Cartesian coordinates, all of the surfaces are planes and they are specified by
each of the independent variables x, y and z separately being a constant.
• In cylindrical coordinates, the surfaces are two planes and a cylinder.
• In spherical coordinates, the surfaces are a sphere, a plane, and a cone.
CARTESIAN COORDINATES
• Coordinates can be written as
or
• The ranges of the variables are:
CIRCULAR CYLINDRICAL COORDINATES
•
The circular cylindrical coordinate system is very
convenient whenever we are dealing with problems
having cylindrical symmetry.
•
A point P in cylindrical coordinates is represented as
:
•
•
•
•
𝜌 is the radius of the cylinder passing through P or the
radial distance from the z-axis:
𝜙, called the azimuthal angle,
and z is the same as in the Cartesian system.
The ranges of the variables are:
FIGURE 1 Point P and unit vectors in the
cylindrical coordinate system.
CIRCULAR CYLINDRICAL COORDINATES
• A vector A in cylindrical coordinates can be written as
or
• Notice that the unit vectors
,
coordinate system is orthogonal.
, and
are mutually perpendicular because our
CIRCULAR CYLINDRICAL COORDINATES
• The relationships between the variables (x, y, z) of
the Cartesian coordinate system and those of the
cylindrical system
are.
• Or
FIGURE 1 Relationship between (x, y, z)
and (ρ, ɸ, z).
CIRCULAR CYLINDRICAL COORDINATES
• The relationships between
• Or
and
are
CIRCULAR CYLINDRICAL COORDINATES
• Finally, the relationships between
and
• Or
𝟏
are
SPHERICAL COORDINATES
• The spherical coordinate system is most appropriate
when dealing with problems having a degree of
spherical symmetry. A point P can be represented as
• A vector A in spherical coordinates can be written as
• or
FIGURE 4 Point P and unit vectors in
spherical coordinates.
SPHERICAL COORDINATES
• The relationships between the variables (x, y, z)
of the Cartesian coordinate system and those of
the cylindrical system
are easily
obtained from.
• Or
FIGURE 5 Relationships between space variables
(x, y, z), (r, θ, ɸ), and (ρ, ɸ, z).
SPHERICAL COORDINATES
• The relationships between
• Or
and
are
DISTANCE BETWEEN TWO POINTS
• The distance between two points is usually necessary in EM theory. The distance d
between two points with position vectors
Cartesian
Cylindrical
spherical
and
is generally given by:
EXERCISE
• Given point P(-2, 6, 3) and vector
, express P and A in
cylindrical and spherical coordinates. Evaluate A at P in the Cartesian, cylindrical,
and spherical systems.
SOLUTION
• At point
• Thus,
. Hence,
SOLUTION
• In the Cartesian system, A at P is
• For A,
• or
. Hence, in the cylindrical system
SOLUTION
•
But
•
At P
, and substituting these yields
𝜌 = 40,
•
6
−2
Hence
cos 𝜙 = −
•
tan 𝜙 =
2
40
,
sin 𝜙 =
6
40
Therefore
𝐴=
−6
40
𝐚 −
38
40
𝐚 = −0.9487𝐚 − 6.008𝐚
SOLUTION
• Similarly, in the spherical system
• Or
SOLUTION
•
But
•
•
At P
•
Hence,
, and substituting these yields
SOLUTION
• Therefore
Note that
is the same in the three systems; that is
EXERCISE
SOLUTION
EXERCISE
• Convert points P(1, 3, 5), T(0, -4, 3), and S(-3, -4, -10) from Cartesian to cylindrical
and spherical coordinates.
• Transform vector to cylindrical and spherical coordinates.
• Evaluate Q at T in the three coordinate systems.
SOLUTION
EXERCISE
• Transform the vector
into cylindrical coordinates.
SOLUTION
EXERCISE
• Express vector
In Cartesian and cylindrical coordinates. Find
and
.
SOLUTION
𝒙
𝒚
𝒛
Or
SOLUTION
But
Hence,
SOLUTION
• Substituting all these gives
SOLUTION
At
Thus,
so
SOLUTION
For spherical to cylindrical vector transformation:
or
SOLUTION
But
Thus,
and
SOLUTION
Hence,
At
, so
Note that |B| is the same in the three systems; that is,
.
This may be used to check the correctness of the result whenever possible.
EXERCISE
• Write an expression for a position vector at any point in space in the rectangular
coordinate system. Then transform the position vector into a vector in the cylindrical
coordinate system.
SOLUTION
CONSTANT-COORDINATE SURFACES
• Surfaces in Cartesian, cylindrical, or spherical
coordinate systems are easily generated by
keeping one of the coordinate variables constant
and allowing the other two to vary.
• In the Cartesian system, if we keep x constant and
allow y and z to vary, an infinite plane is
generated. Thus we could have infinite planes
FIGURE 2.7 Constant x, y, and z surfaces.
which are perpendicular to the x-, y-, and z-axes,
respectively, as shown in Figure.
CONSTANT-COORDINATE SURFACES
• The intersection of two planes is a line. For
example,
RPQ parallel to the z-axis.
is the line
• The intersection of three planes is a point. For
• example,
is the point P(x, y, z).
• Thus we may define point P as the intersection of
three orthogonal infinite planes. If P is (1, -5, 3),
then P is the intersection of planes x = 1, y = -5,
and z = 3.
FIGURE 2.7 Constant x, y, and z surfaces.
CONSTANT-COORDINATE SURFACES
• Orthogonal surfaces in cylindrical coordinates can
likewise be generated. The surfaces
are illustrated in Figure 2.8, where it is easy to
observe that
constant is a circular cylinder,
= constant is a semi-infinite plane with its edge
along the z-axis, and z = constant is the same infinite
plane as in a Cartesian system.
FIGURE 2.8 Constant ρ, ɸ, and z surfaces.
CONSTANT-COORDINATE SURFACES
• Where two surfaces meet is either a line or a
circle. Thus,
is a
circle QPR of radius p, whereas z = constant,
constant is a semi-infinite line.
• A point is an intersection of the three surfaces.
Thus, p = 2,
5).
= 60°, z = 5 is the point P(2, 60°,
FIGURE 2.8 Constant ρ, ɸ, and z surfaces.
CONSTANT-COORDINATE SURFACES
• The orthogonal nature of the spherical coordinate
system is evident by considering the three surfaces
which are shown in Figure 2.9, where we notice that
r=constant is a sphere with its center at the origin;
=constant is a circular cone with the z-axis as its
axis and the origin as its vertex;
constant is the
semi-infinite plane as in a cylindrical system.
FIGURE 2.9 Constant r, θ, and ɸ
surfaces.
CONSTANT-COORDINATE SURFACES
• A line is formed by the intersection of two surfaces.
For example:
is a
semicircle passing through Q and P. The intersection
of three surfaces gives a point.
• Thus, r = 5,
30°,
60° is the point P(5, 30°,
60°).
• We notice that in general, a point in threedimensional space can be identified as the
intersection of three mutually orthogonal surfaces.
FIGURE 2.9 Constant r, θ, and ɸ
surfaces.
CONSTANT-COORDINATE SURFACES
• Also, a unit normal vector to the surface n =
constant is
.
, where n is x, y, z, , , r, or
• For example, to plane x=5, a unit normal
vector is
and to planed
normal vector is
=20°, a unit
.
FIGURE 2.9 Constant r, θ, and ɸ
surfaces.
EXERCISE
• Two uniform vector fields are given by:
•
and
•
• The vector component of E at
• The angle E makes with the surface
. Calculate
parallel to the line
at P.
.
SOLUTION
• 1.
𝐚
𝐚
𝐚
𝐄 × 𝐅 = −5 10 3
1
2 −6
= −66, −27, −20
𝐄×𝐅 =
66 + 27 + 20 = 74.06
• 2.
• Line 𝑥 = 2, 𝑧 = 3 is parallel to y-axis, so the component of E parallel to the given line is
𝐄. 𝐚 𝐚
But at 𝑃(5, 𝜋⁄2 , 3)
𝐚 = sin 𝜙 𝐚 + cos 𝜙 𝐚
= sin 𝜋⁄2 𝐚 + cos 𝜋⁄2 𝐚 = 𝐚
• Therefore,
𝐄. 𝐚 𝐚 = 𝐄. 𝐚 𝐚 = −5𝐚
SOLUTION
• 3. Utilizing the fact that the z-axis is normal to the surface
, we can use the dot
product to find the angle between the z-axis and E, as shown in Figure.
•
•
• Hence, the angle between
•
and E is