Page 36 - Engineering Electromagnetics, 8th Edition

18 ENGINEERING ELECTROMAGNETICS

EXAMPLE 1.3
Transform the vector B = ya x − xa y + za z into cylindrical coordinates.

Solution. The new components are
B ρ = B · a ρ = y(a x · a ρ ) − x(a y · a ρ )
= y cos φ − x sin φ = ρ sin φ cos φ − ρ cos φ sin φ = 0
B φ = B · a φ = y(a x · a φ ) − x(a y · a φ )
2
2
=−y sin φ − x cos φ =−ρ sin φ − ρ cos φ =−ρ
Thus,

B =−ρa φ + za z

D1.5. (a)Give the rectangular coordinates of the point C(ρ = 4.4,φ =
◦
−115 , z = 2). (b)Give the cylindrical coordinates of the point D(x =
−3.1, y = 2.6, z =−3). (c) Specify the distance from C to D.
Ans. C(x =−1.860, y =−3.99, z = 2); D(ρ = 4.05,φ = 140.0 , z =−3); 8.36
◦

D1.6. Transform to cylindrical coordinates: (a) F = 10a x −8a y +6a z at point
P(10, −8, 6); (b) G = (2x + y)a x −(y −4x)a y at point Q(ρ, φ, z). (c)Give the
rectangular components of the vector H = 20a ρ − 10a φ + 3a z at P(x = 5,
y = 2, z =−1).

2 2 2 2
Ans. 12.81a ρ +6a z ;(2ρ cos φ −ρ sin φ +5ρ sin φ cos φ)a ρ +(4ρ cos φ −ρ sin φ
− 3ρ sin φ cos φ)a φ ; H x = 22.3, H y =−1.857, H z = 3

1.9 THE SPHERICAL COORDINATE SYSTEM
We have no two-dimensional coordinate system to help us understand the three-
dimensional spherical coordinate system, as we have for the circular cylindrical
coordinate system. In certain respects we can draw on our knowledge of the latitude-
and-longitude system of locating a place on the surface of the earth, but usually we
consider only points on the surface and not those below or above ground.
Let us start by building a spherical coordinate system on the three rectangular
axes (Figure 1.8a). We first define the distance from the origin to any point as r. The
surface r = constant is a sphere.
The second coordinate is an angle θ between the z axis and the line drawn
from the origin to the point in question. The surface θ = constant is a cone, and
the two surfaces, cone and sphere, are everywhere perpendicular along their inter-
section, which is a circle of radius r sin θ. The coordinate θ corresponds to latitude,