cksoon11
Nov15-09, 10:02 PM
1. The problem statement, all variables and given/known data
For a harmonic uniform plane wave propagating in a simple medium, both \vec{E} and \vec{H} vary in accordance with the factor exp(-i \vec{k}.\vec{R})
Show that the four Maxwell’s equations
for a uniform plane wave in a source-free region reduce to the following:
\vec{k} \times\vec{E}= \omega\mu\vec{H}
\vec{k} \times\vec{H} = \omega\epsilon\vec{E}
\vec{k} \bullet \vec{E} = 0
\vec{k} \bullet \vec{H} = 0
Apparently "Vector k and Vector R are the the general forms of wave number and position vector(or direction of propagation)"
Question Source(no.4) :http://www.lib.yuntech.edu.tw/exam_new/96/de.pdf
2. Relevant equations
You supposed to use Maxwell's Equations for a plane wave
\nabla x \vec{E} = -i\omega\mu\vec{H}
\nabla x \vec{H} = i\omega\mu\vec{E}
\nabla . \vec{E} = 0
\nabla . \vec{H} = 0
3. The attempt at a solution
First off,I am confused as to how k can be a vector when it is the wave number(a scalar).
From what I can tell,the wave is propagating in the radial direction in spherical coordinates.
I then assumed the electric and magnetic fields to be orthogonal in the \thetaand \phi direction.
But just simply substituting the phasor form of the plane wave into Maxwell's equations:
E_{o} exp((-i \vec{k}.\vec{R}) into Maxwell equations doesn't seem to yield the desired results because I don't understand how they obtained the cross-products and dot products of \vec{k}with \vec{E} and
\vec{H}.
I just can't seem to grasp the concept of a vector as my wave number.Could someone please phrase the question in more concise terms?What am I misunderstanding here?
For a harmonic uniform plane wave propagating in a simple medium, both \vec{E} and \vec{H} vary in accordance with the factor exp(-i \vec{k}.\vec{R})
Show that the four Maxwell’s equations
for a uniform plane wave in a source-free region reduce to the following:
\vec{k} \times\vec{E}= \omega\mu\vec{H}
\vec{k} \times\vec{H} = \omega\epsilon\vec{E}
\vec{k} \bullet \vec{E} = 0
\vec{k} \bullet \vec{H} = 0
Apparently "Vector k and Vector R are the the general forms of wave number and position vector(or direction of propagation)"
Question Source(no.4) :http://www.lib.yuntech.edu.tw/exam_new/96/de.pdf
2. Relevant equations
You supposed to use Maxwell's Equations for a plane wave
\nabla x \vec{E} = -i\omega\mu\vec{H}
\nabla x \vec{H} = i\omega\mu\vec{E}
\nabla . \vec{E} = 0
\nabla . \vec{H} = 0
3. The attempt at a solution
First off,I am confused as to how k can be a vector when it is the wave number(a scalar).
From what I can tell,the wave is propagating in the radial direction in spherical coordinates.
I then assumed the electric and magnetic fields to be orthogonal in the \thetaand \phi direction.
But just simply substituting the phasor form of the plane wave into Maxwell's equations:
E_{o} exp((-i \vec{k}.\vec{R}) into Maxwell equations doesn't seem to yield the desired results because I don't understand how they obtained the cross-products and dot products of \vec{k}with \vec{E} and
\vec{H}.
I just can't seem to grasp the concept of a vector as my wave number.Could someone please phrase the question in more concise terms?What am I misunderstanding here?