Page 34 - Engineering Electromagnetics, 8th Edition

16 ENGINEERING ELECTROMAGNETICS

Figure 1.7 The relationship between
the rectangular variables x, y, z and the
cylindrical coordinate variables ρ, φ, z.
There is no change in the variable z
between the two systems.

The variables of the rectangular and cylindrical coordinate systems are easily
related to each other. Referring to Figure 1.7, we see that

x = ρ cos φ
y = ρ sin φ (10)
z = z

From the other viewpoint, we may express the cylindrical variables in terms of x, y,
and z:

2
ρ = x + y 2 (ρ ≥ 0)
y
φ = tan −1 (11)
x
z = z
We consider the variable ρ to be positive or zero, thus using only the positive sign
for the radical in (11). The proper value of the angle φ is determined by inspecting
the signs of x and y. Thus, if x =−3 and y = 4, we find that the point lies in the
second quadrant so that ρ = 5 and φ = 126.9 .For x = 3 and y =−4, we have
◦
φ =−53.1 or 306.9 , whichever is more convenient.
◦
◦
Using (10) or (11), scalar functions given in one coordinate system are easily
transformed into the other system.
Avector function in one coordinate system, however, requires two steps in order
to transform it to another coordinate system, because a different set of component