How Does an Electromagnetic Wave Propagate in Glass?

In summary: I am still not sure why I was supposed to assume that ##\mu = \mu_0##. I thought it was a special case. But since we're talking about glasses here, I guess it is.
  • #1
TheSodesa
224
7

Homework Statement



A harmonic EM-wave is propagating in glass in the +x-direction. The refractive index of the glass ##n = 1.4##. The wave number of the wave ##k = 30 \ rad/m##. The magnetic portion of the wave is parallel to the y-axis and its amplitude ##H_0 = 0.10A/m##. At ##t=0## and ##x = 0## the magnetic field vector ##\vec{H} = H_0 \hat{j} = +0.10A/m \ \hat{j}##.

a) Approximate the values for the permittivity (##\epsilon##) and permeability (##\mu##) of the type of glass in question.
b) Calculate the propagation speed, wavelength, angular frequency and frequency of the wave.
c) Calculate the amplitude of the electric field component of the wave at x=0, t=0 (remember to show the direction of the vector).
d) Write the expressions for the electric and magnetic field components in vector form.

Homework Equations

Basic wave equation:
\begin{equation}
v = \lambda f
\end{equation}
The wave number:
\begin{equation}
k = \frac{\omega}{v} = \frac{2\pi f}{v}
\end{equation}
Relation between the magnetic and electric components of the wave resulting from them being in phase:
\begin{equation}
E = vB
\end{equation}
The definition of the refractive index:
\begin{equation}
n:= \frac{c}{v} \stackrel{Maxwell}{=} \frac{\sqrt{\epsilon \mu}}{\sqrt{\epsilon_0 \mu_0}}
\end{equation}
Total energy density of an EM-wave:
\begin{equation}
w = w_E + w_M = \frac{\epsilon E^2}{2} + \frac{\mu H^2}{2}
\end{equation}
As a result of ##E = vB##, the ratio of the energy densities of the electric and magnetic field components:
\begin{equation}
\frac{w_E}{w_M} = \mu \epsilon v^2 = 1
\end{equation}

The Attempt at a Solution



a) I'm having trouble with this part. Looking at equations (4) and (6):
\begin{cases}
\epsilon \mu &= n^2 \epsilon_0 \mu_0\\
\epsilon \mu &= \frac{1}{v^2}
\end{cases}
This should be easily solvable, but my math head is not working(again). If I solve for either ##\epsilon## or ##\mu## in one equation and plug it into the other equation, it will cancel out and I'm only left with things that I already have values for. It seems I need more information. Not sure what.

b) Looking at equation (3) the velocity:
[tex]v = \frac{c}{n} = \frac{2.9979 \times 10^{8}m/s}{1.4} = 214 \ 135 \ 714.3 \ m/s[/tex]
Now from equation (1) the angular frequency:
\begin{array}{lll}
\omega
&= kv \\
&= (30rad/m)(214 \ 135 \ 714.3 \ m/s)\\
&= 6 \ 424 \ 071 \ 429 \ rad/s\\
&\approx 6.4 \ G \ rad/s
\end{array}
and frequency:
\begin{array}{lll}
f
&= \frac{\omega}{2 \pi}\\
&= \frac{6 \ 424 \ 071 \ 429 \ rad/s}{2\pi}\\
&= 1 \ 022 \ 422 \ 723 \ Hz\\
&\approx 1.0 \ GHz
\end{array}
Equation (1) gives us the wavelength:
\begin{array}{ll}
\lambda
&= \frac{c}{f}\\
&= \frac{214 \ 135 \ 714.3 \ m/s}{1 \ 022 \ 422 \ 723 \ Hz}\\
&= 0.2 \ 094 \ 395 \ 102 \ m\\
&\approx 0.21 \ m
\end{array}

c) From equation (3):

\begin{array}{ll}
E_0
&= vB_0\\
&= v \mu H_0
\end{array}
In need the value of ##\mu## for this (from part a). The direction can be found out by using the right hand rule. If ##\hat{H}## is in the y-direction and the wave is propagating in the x-direction, the electric component must be in the z- direction. Since the waves are in phase, ##\hat{E}## must be in the +z-direction.(Right?)

d) Since the wave is harmonic, its components are sinusoidal. From c we know the directions of the components, so we can write:
\begin{array}
\vec{E} = E_0 sin(\omega t - kx) \hat{k}\\
\vec{H} = H_0 sin(\omega t - kx) \hat{j}
\end{array}

And there you have it. All I really need help with is part a. After that I should be able to solve for the electric field in part c. I assume the rest of my answer is correct.
 
Last edited:
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  • #2
To actually show some work in part a):

\begin{cases}
\epsilon \mu &= n^2 \epsilon_0 \mu_0\\
\epsilon \mu &= \frac{1}{v^2}
\end{cases}

\begin{cases}
\mu &=\frac{n^2 \epsilon_0 \mu_0}{\epsilon }\\
\epsilon &= \frac{1}{ \mu v^2}
\end{cases}

\begin{cases}
\mu &=\frac{n^2 \epsilon_0 \mu_0}{\frac{1}{ \mu v^2}} = n^2 \epsilon_0 \mu_0 \mu v^2|| \ \text{Divide by } \mu\\
\epsilon &= \frac{1}{ \mu v^2}
\end{cases}

\begin{cases}
1 &= n^2 \epsilon_0 \mu_0 v^2\\
\epsilon &= \frac{1}{ \mu v^2}
\end{cases}
This would allow me to solve for ##v##, but I've already done that by other means... It's as if equations (4) and (6) were the same equation.
 
  • #3
TheSodesa said:
To actually show some work in part a):
\begin{cases}
\epsilon \mu &= n^2 \epsilon_0 \mu_0\\
\end{cases}
Why don't you just solve for μ from this?
Everything else looks OK to me.
 
  • #4
rude man said:
Why don't you just solve for μ from this?
Everything else looks OK to me.

I'm a bit late in answering this, but it turned out I was supposed to assume that the permeability of glass, ##\mu##, is equal to ##\mu _0##. From there on the question solved itself, except part d. In part d I forgot to add the phase angle of ##\frac{\pi}{2}## inside of the sines as follows:
\begin{cases}
E = E_0 sin(\omega t - kx + \frac{\pi}}{2}) \hat{k}\\
H = H_0 sin(\omega t - kx + \frac{\pi}}{2}) \hat{j}
\end{cases}
since at ##t=0## and ##x=0##, ##H=H_0##. Also the sign of E was brought into question by the assistants in part c. If I had used the traditional right handed coordinate system, ##E## would have been negative at ##t=0## and ##x=0##.

In any case, I will mark this as solved.
 

Related to How Does an Electromagnetic Wave Propagate in Glass?

1. What is an electromagnetic wave in glass?

An electromagnetic wave in glass is a type of wave that consists of electric and magnetic fields oscillating perpendicular to each other as it travels through glass. These waves are also known as light waves or electromagnetic radiation.

2. How do electromagnetic waves travel through glass?

Electromagnetic waves travel through glass by passing through the atoms and molecules of the glass material. As the wave travels through the glass, the electric and magnetic fields interact with the charged particles in the glass, causing the wave to slow down.

3. What is the speed of an electromagnetic wave in glass?

The speed of an electromagnetic wave in glass is slower than the speed of light in a vacuum, which is approximately 299,792,458 meters per second. The exact speed depends on the composition of the glass and the wavelength of the wave.

4. What is the relationship between the refractive index of glass and electromagnetic waves?

The refractive index of glass is a measure of how much the speed of light is slowed down when passing through the glass. This means that the higher the refractive index, the slower the speed of electromagnetic waves in glass.

5. Can electromagnetic waves be absorbed by glass?

Yes, electromagnetic waves can be absorbed by glass. When an electromagnetic wave encounters a molecule in the glass, the energy of the wave can be transferred to the molecule, causing it to vibrate and heat up. This absorption of energy results in the wave losing intensity as it passes through the glass.

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