Page 532 - Engineering Electromagnetics, 8th Edition

514 ENGINEERING ELECTROMAGNETICS

µI 0 d
A θs =− sin θ e − jkr (7b)
4πr
From these two components of the vector magnetic potential at P we can now
find B s or H s from the definition of A s ,
(8)
B s = µH s =∇ × A s
Taking the indicated partial derivatives as specified by the curl operator in spherical
coordinates, we are able to separate Eq. (8) into its three spherical components, of
which only the φ component is non-zero:
1 ∂ 1 ∂ A rs
H φs = (rA θs ) − (9)
µr ∂r µr ∂θ
Now, substituting (7a) and (7b) into (9), we find the magnetic field:

I 0 d k 1
H φs = sin θ e − jkr j + (10)
4π r r 2
The electric field that is associated with Eq. (10) is found from one of Maxwell’s
equations—specifically the point form of Ampere’s circuital law as applied to the
surrounding region (where conduction and convection current are absent). In phasor
form, this is Eq. (23) in Chapter 11, except that in the present case we allow for a
lossless medium having permittivity :

∇× H s = jω E s (11)
Using (11), we expand the curl in spherical coordinates, assuming the existence of
only a φ component for H s . The resulting electric field components are:
1 1 ∂
E rs = (H φs sin θ) (12a)
jω r sin θ ∂θ
1 1 ∂
E θs = − (rH φs ) (12b)
jω r ∂r
Then on substituting (10) into (12a) and (12b) we find:

I 0 d 1 1
E rs = η cos θ e − jkr 2 + 3 (13a)
2π r jkr
I 0 d − jkr jk 1 1
E θs = η sin θ e + + (13b)
4π r r 2 jkr 3
√
where the intrinsic impedance is, as always, η = µ/ .
Equations (10), (13a), and (13b) are the fields that we set out to find. The next step
is to interpret them. We first notice the e − jkr factor appearing with each component.
By itself, this term describes a spherical wave, propagating outward from the origin
in the positive r direction with a phase constant k = 2π/λ. λ is the wavelength as