Page 38 - Engineering Electromagnetics, 8th Edition

20 ENGINEERING ELECTROMAGNETICS

direction in which the coordinate increases. The unit vector a r is directed radially
outward, normal to the sphere r = constant, and lies in the cone θ = constant and
the plane φ = constant. The unit vector a θ is normal to the conical surface, lies in
the plane, and is tangent to the sphere. It is directed along a line of “longitude” and
points “south.” The third unit vector a φ is the same as in cylindrical coordinates, being
normal to the plane and tangent to both the cone and the sphere. It is directed to the
“east.”
The three unit vectors are shown in Figure 1.8c. They are, of course, mutually per-
pendicular, and a right-handed coordinate system is defined by causing a r × a θ = a φ .
Our system is right-handed, as an inspection of Figure 1.8c will show, on application
of the definition of the cross product. The right-hand rule identifies the thumb, fore-
finger, and middle finger with the direction of increasing r, θ, and φ, respectively.
(Note that the identification in cylindrical coordinates was with ρ, φ, and z, and in
rectangular coordinates with x, y, and z.) A differential volume element may be con-
structed in spherical coordinates by increasing r, θ, and φ by dr, dθ, and dφ,as
shown in Figure 1.8d. The distance between the two spherical surfaces of radius r
and r + dr is dr; the distance between the two cones having generating angles of θ
and θ + dθ is rdθ; and the distance between the two radial planes at angles φ and
φ + dφ is found to be r sin θdφ, after a few moments of trigonometric thought. The
2
surfaces have areas of rdrdθ, r sin θ dr dφ, and r sin θ dθ dφ, and the volume is
2
r sin θ dr dθ dφ.
The transformation of scalars from the rectangular to the spherical coordinate
system is easily made by using Figure 1.8a to relate the two sets of variables:

x = r sin θ cos φ
y = r sin θ sin φ (15)
z = r cos θ

The transformation in the reverse direction is achieved with the help of

2
2
r = x + y + z 2 (r ≥ 0)
z
θ = cos −1 (0 ≤ θ ≤ 180 ) (16)
◦
◦

x + y + z 2
2
2
y
φ = tan −1
x
The radius variable r is nonnegative, and θ is restricted to the range from 0 to 180 ,
◦
◦
inclusive. The angles are placed in the proper quadrants by inspecting the signs of
x, y, and z.
The transformation of vectors requires us to determine the products of the unit
vectors in rectangular and spherical coordinates. We work out these products from
Figure 1.8c and a pinch of trigonometry. Because the dot product of any spheri-
cal unit vector with any rectangular unit vector is the component of the spherical