8/19/2019 Typical Maxwell's Equation Problem With Solution
1/2
Typical Maxwell’s Equation Problem with Solution
1. The wave equations (Helmhotlz equations) for the
electric and magnetic field intensity are given by:
⃗ ⃗ (1) ⃗ ⃗ (2) where the electric field
has only an x component and the fields exists in a material
mediumcharacterized by the permittivity (ε) and permeability
( μ). For a lossless medium and with the notationk
2 = ω
2 με, the general solution of (1) is:
⃗ — (3) a) Write the expression of the
electric field intensity in (3) in time domain, i.e.,
considering
cosine-time dependence. b) Demonstrate that the
propagation constant k is equal also with k =
2π/ λ.c) Calculate the associated magnetic field
intensity vector using the appropriate Maxwell’s curl
equation. Consider only the + z -direction travelling
electric wave. Specify the magnetic field
amplitude as a function of E 0+ and find its
direction. (Hint: start with the matrix form of the
curl equation).d) By inspection of (4) and calculations,
find the amplitude E 0, frequency (GHz), the
propagation constant (k ), the phase velocity
(v p), the dielectric constant εr and the
waveimpedance (η) of the wave described by the electric field in
(4). The wave propagate inforward direction in a nonmagnetic
medium, i.e., μr = 1. You know η0 = 377
Ω.
⃗ (4)
2. Solution:
a)
⃗ — (3)⃗ (S1)
b) Demonstrate k = 2π/ λ
√ (S2)c)
⃗ ⃗ ⃗ (S3)
⃗ |
| |
|
(
—)
—(S4)
From (S3) and (S4)
