Chapter 31, Plane Electromagnetic Waves Iv Video Solutions, Electromagnetic Fields and Waves: Including Electric Circuits

Problem 1

(31.1) A nonuniform plane electromagnetic wave is not transverse.
We define a transverse wave as one in which $\boldsymbol{E}$ and $\boldsymbol{H}$ are both perpendicular to the two vectors $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$.
(a) Write out Maxwell's equations for a plane sinusoidal wave in free space, replacing $\boldsymbol{V}$ by $j \boldsymbol{k}$ and $\partial / \partial t$ by $j \omega .$ Note that $\boldsymbol{k} \cdot \boldsymbol{E}$ is always equal to zero. Since the vector $\boldsymbol{k}$ is complex, it has no specific orientation in space, except that it lies in the plane defined by the vectors $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$.
(b) Suppose that $\boldsymbol{H}$ is transverse: $\boldsymbol{H}=\mathrm{H} \hat{\boldsymbol{n}}$, where $\hat{\boldsymbol{n}}+\boldsymbol{\alpha}=0$ and $\hat{\boldsymbol{n}} \cdot \boldsymbol{\beta}=0$. Show that $\boldsymbol{E}$ is then not transverse. Similarly, if $\boldsymbol{E}$ is transverse, then $\boldsymbol{H}$ is not transverse.

Zachary Warner

Numerade Educator

02:41

Problem 2

(31.2) Total reflection as in Fig. $31-5$
An electromagnetic wave polarized with its $\boldsymbol{E}$ vector normal to the plane of incidence is totally reflected as in Fig. $31-5$ at the interface between a dielectric whose index of refraction is $3.0$ and air. The angle of incidence is $75^{\circ}$
(a) Calculate $\delta_{z} / \lambda_{2}$ and $\delta_{z} / \lambda_{1}$
(b) Calculate the phases of the reflected and transmitted waves with respect to the incident wave at any point on the interface.
(c) Check the continuity of $\boldsymbol{E}$ across the interface.

Mohamed Raafat Mohamed

Numerade Educator

05:45

Problem 3

(31.2) The Poynting vector for the transmitted wave Check the value of $\mathscr{Y}_{T, \text { av } \perp \text { given in }}$ Sec. $31.2 .2$, for $n_{2}=1$.

Laszlo Zalavari

Numerade Educator

03:50

Problem 4

(31.2) Total reflection on a plasma when $\omega / \omega_{p}<1$ )
Show that a wave incident on an ionized region is totally reflected if $\omega<\omega_{p}$, where $\omega_{p}$ is the plasma angular frequency (Sec. 29.2.3).

Melissa Munoz

Numerade Educator

05:38

Problem 5

31-5. (31.2) The phase shifts $\Phi_{!}$and $\Phi_{\perp}$ in total reflection
(a) Show that
$$
\Phi_{h}=-2 \arctan \frac{\cos \theta_{I}}{\left(n_{1} / n_{2}\right)^{2}\left(\sin ^{2} \theta_{T}-n_{2}^{2} / n_{1}^{2}\right)^{1 / 2}}
$$
(b) Show that
$$
\tan \left(\frac{\Phi_{\perp}}{2}-\frac{\Phi_{1}}{2}\right)=\frac{\sin ^{2} \theta_{I}}{\cos \theta_{l}\left(\sin ^{2} \theta_{l}-n_{2}^{2} / n_{1}^{2}\right)^{1 / 2}}
$$
(c) Plot $\Phi_{1}-\Phi_{\|}$as a function of $\theta_{i}$ between $40^{\circ}$ and $90^{\circ}$ for $n_{1}=1.5$ and $n_{2}=1$

Khoobchandra Agrawal

Numerade Educator

01:49

Problem 6

31-6. (31.2) The transmitted wave when $\boldsymbol{E}$ is parallel to the plane of incidence Show that
$$
\left(\frac{E_{T m}}{E_{l m}}\right)_{\|}=\frac{2}{\left[\left(n_{2} / n_{1}\right)^{2}+1\right]^{1 / 2}} \exp j\left(\frac{\Phi_{1}}{2}-\frac{\pi}{2}\right)
$$

Suzanne W.

Numerade Educator

02:56

Problem 7

(31.2) Scintillation particle detector
Figure $31-10$ shows one type of scintillation particle detector. A scintillator $S$, usually made out of a single crystal of sodium iodide or of a suitable transparent plastic embedded in a reflector $R$, emits light when it is traversed by an ionizing particle such as an electron. A photomultiplier PM detects the emitted light.

The scintillator has an index of refraction $n_{1}$ and is fixed to the face of the photomultiplier with a cement $C$ of index $n_{2}<n_{1}$. Light is emitted in all directions in the scintillator, but only a fraction $F$ reaches the photomultiplier.
(a) Calculate $F$ as a function of $n_{1} / n_{2}$, assuming that $T=1$ for angles of incidence smaller than the critical angle and that the scintillator is surrounded by a nonreflecting substance.
(b) Draw a graph of $F$ for values of $n_{2} / n_{1}$ ranging from $0.1$ to $1.0$.

Manish Jain

Numerade Educator

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Problem 8

(31.2) Total reflection in light-emitting diodes
In light-emitting diodes (LEDs), radiation occurs in a junction plane within a semiconductor whose index of refraction is quite large. For example, with GaAsP, $n=3.5$. Total reflection at the semiconductor-air interface limits the efficiency of LEDs to a few percent.
(a) Calculate the critical angle.
(b) Plot curves of $T_{1}$ and $T_{\perp}$ for $n_{1} / n_{2}=3.5$. Note that the transmission coefficients are equal and approximately independent of the angle of incidence when $\theta_{l}$ is small. Identify the Brewster angle.
(c) Assume that the face of the semiconductor is flat and parallel to the junction. The index of refraction is $n$.

Calculate the fraction $F$ of the light emitted at the source that reaches the surface at an angle smaller than the critical angle. Show that $F \approx 1 /\left(4 n^{2}\right)$.
(d) Show that $F T=1 /\left[n(n+1)^{2}\right]$.
(e) Calculate $F, T$, and $F T$ for $n=3.5$.
(f) Calculate $F T$ for an LED situated at the center of a hemisphere whose index of refraction is the same as that of the semiconductor. This is impractical because shaping the semiconductor is expensive.
(g) LEDs are usually covered with a hemispherical transparent resin whose $n$ is about $1.6 .$

Calculate the two transmission coefficients and the efficiency. The efficiency is improved, but it is still very low.

Lainey Roebuck

Numerade Educator

01:29

Problem 9

(31.2) The $\boldsymbol{H}$ vector of the transmitted wave
We found that the $x$ and $z$ components of the $H_{T}$ vector of the transmitted wave are in quadrature when there is total reflection with the $\boldsymbol{E}$ vector normal to the plane of incidence.
Show that the vector rotates in the direction shown in Fig. 31-6.

Ajay Singhal

Numerade Educator