Chapter 9: Q9.30P (page 430)

Confirm that the energy in the $T E_{m n}$ mode travels at the group velocity. [Hint: Find the time-averaged Poynting vector $< S >$ and the energy density $< u >$ (use Prob. 9.12 if you wish). Integrate over the cross-section of the waveguide to get the energy per unit time and per unit length carried by the wave, and take their ratio.]

Short Answer

Expert verified

Answer

The velocity of energy carried by the electromagnetic wave is confirmed to be the group velocity of the wave.

Step by step solution

Expression for the velocity of the energy carried by the electromagnetic wave traveling in a waveguide:

Write the expression for the velocity of the energy carried by the electromagnetic wave traveling in a waveguide.

$V = \frac{\int < \vec{S} > d \vec{A}}{\int < \vec{u} > d \vec{A}}$ …… (1)

Here, v is the velocity of electromagnetic wave, $<\vec{S}>$ is the time-averaged pointing vector for the electromagnetic wave, $<\vec{u}>$ is the energy per unit volume of the electromagnetic wave, and $\vec{A}$ is the cross-section of the waveguide.

Write the expression for the time-averaged pointing vector for the electromagnetic wave.

$<S> = \frac{1}{2 μ_{0}} R e (\vec{E} \times \vec{B})$ …… (2)

Write the expression for the energy per unit volume of the electromagnetic wave.

$<u> = \frac{1}{4} R e (ε_{0} \vec{E} \vec{E} + \frac{1}{μ_{0}} \vec{B} \vec{B})$ …… (3)

Here, $*$ represents the complex conjugate of the quantity.

Determine the time-averaged pointing vector for the electromagnetic wave:

Write the value of $(\vec{E} \times \vec{B})$ .

$(\vec{B} \times \vec{B}) = (B_{z}^{*} E_{y}) \hat{x} - (B_{z}^{*} E_{x}) \hat{x} + (B_{z}^{*} E_{x} - B_{z}^{*} E_{y}) \hat{z}$

As $(B_{z}^{*} E_{y})$ and $(B_{z}^{*} E_{x})$ are imaginary components, the value of localid="1658406246663" $R e (\vec{E} \times \vec{B})$ will be,

localid="1658406255214" $R e (\vec{E} \times \vec{B}) = (B_{z}^{*} E_{x} - B_{z}^{*} E_{y}) \hat{z}$

Using equation 9.180 and 9.186, write the value of $B_{y}^{*}, E_{x}, B_{x}^{*}$ and $E_{y}$ .

$B_{y}^{*} = \frac{- i k}{{(\frac{ω}{c})}^{2} - k^{2}} (- \frac{n π}{b}) B_{0} \cos (\frac{m π x}{a}) \sin (\frac{n π y}{b}) E_{x} = \frac{- iω}{{(\frac{ω}{c})}^{2} - k^{2}} (\frac{- nπ}{b}) B_{0} \cos (\frac{mπx}{a}) \sin (\frac{nπy}{b}) B_{x}^{*} = \frac{- ik}{{(\frac{ω}{c})}^{2} - k^{2}} (\frac{mπ}{a}) B_{0} \sin (\frac{mπx}{a}) \cos (\frac{nπy}{b}) E_{y} = \frac{- ik}{{(\frac{ω}{c})}^{2} - k^{2}} (\frac{- mπ}{a}) B_{0} \sin (\frac{mπx}{a}) \cos (\frac{nπy}{b})$

Substitute all the above values in equation (2).

$< S > = \frac{1}{2 μ_{0}} (B_{y}^{*} E_{x} - B_{x}^{*} E_{y}) \hat{z} < S > = \frac{ω k π^{2} B_{0}^{2}}{{(\frac{ω}{c})}^{2} - k^{2}} [\begin{matrix} {(\frac{n}{b})}^{2} c o s^{2} (\frac{m π x}{a}) s i n^{2} (\frac{n π y}{b}) + \\ {(\frac{m}{a})}^{2} s i n^{2} (\frac{m π x}{a}) c o s^{2} (\frac{n π y}{b}) \hat{z} \end{matrix}] \int < S > \cdot d a = \frac{1}{8 μ_{0}} \frac{ω k π^{2} B_{0}^{2}}{{(\frac{ω}{c})}^{2} - k^{2}} a b [{(\frac{m}{a})}^{2} + {(\frac{n}{b})}^{2}] \int < S > \cdot d a = \frac{1}{8 μ_{0}} \frac{ω k c^{2} B_{0}^{2} a b}{(ω_{m n}^{2})}$

Determine the energy per unit volume of the electromagnetic wave:

Using the equation9.176, write the value of $\vec{E}$ and $\vec{B}$ .

$\vec{E} = {\vec{E}}_{0} e^{i (kz - ωt)} {\vec{B}}^{*} = {\vec{B}}^{*}_{0} e^{i (kz - ωt)}$

Similarly, using equation 9.180 and 9.186, write the value of $B_{z}^{*}$ and $E_{z}$ .

$B_{z}^{*} = B_{0} \cos (\frac{mπx}{a}) \cos (\frac{nπy}{b}) E_{z} = 0$

Substitute the value of localid="1658406358582" $\vec{E}$ and localid="1658406367803" $\vec{B}$ in equation (3).

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#9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage(' $" localid="1658406400397" < u > = \frac{1}{4} R e (ε_{0} \overset{}{E} \overset{}{E} + \frac{1}{μ_{0}} \overset{}{B} \overset{}{B}) < u > = \frac{ε_{0}}{4} \frac{ω^{2} π^{2} B_{0}^{2}}{{(\frac{ω}{c})}^{2} - k^{2}} {\begin{matrix} [\begin{matrix} {(\frac{n}{b})}^{2} c o s^{2} (\frac{m π x}{a}) s i n^{2} (\frac{n πy}{b}) + \\ {(\frac{m}{a})}^{2} s i n^{2} (\frac{m π x}{a}) c o s^{2} (\frac{n π y}{b}) \end{matrix}] + \\ \frac{1}{4 μ_{0}} [\begin{matrix} B_{0}^{2} c o s^{2} (\frac{m π x}{a}) c o s^{2} (\frac{n π y}{b}) + \\ \frac{< u > = \frac{1}{4} Re (ε_{0} \overset{}{E} \overset{}{E} + \frac{1}{μ_{0}} \overset{}{B} \overset{}{B}) < u > = \frac{ε_{0}}{4} \frac{ω^{2} π^{2} B_{0}^{2}}{{(\frac{ω}{c})}^{2} - k^{2}} {\begin{matrix} [\begin{matrix} {(\frac{n}{b})}^{2} \cos^{2} (\frac{mπx}{a}) \sin^{2} (\frac{nπy}{b}) + \\ {(\frac{m}{a})}^{2} \sin^{2} (\frac{mπx}{a}) \cos^{2} (\frac{nπy}{b}) \end{matrix}] + \\ \frac{1}{4 μ_{0}} [\begin{matrix} B_{0}^{2} \cos^{2} (\frac{mπx}{a}) \cos^{2} (\frac{nπy}{b}) + \\ \frac{k^{2} π^{2} B_{0}^{2}}{{(\frac{ω}{c})}^{2} - k^{2}} [\begin{matrix} {(\frac{n}{b})}^{2} \cos^{2} (\frac{mπx}{a}) \\ \sin^{2} (\frac{nπy}{b}) + \\ {(\frac{m}{a})}^{2} \sin^{2} (\frac{mπx}{a}) \cos^{2} (\frac{nπy}{b}) \end{matrix}] \end{matrix}] \end{matrix}} \int < u > \cdot da = \frac{ab}{4} {\frac{ε_{0}}{4} \frac{ω^{2} π^{2} B_{0}^{2}}{{(\frac{ω}{c})}^{2} - k^{2}} [{(\frac{n}{b})}^{2} - {(\frac{m}{a})}^{2}] + \frac{B_{0}^{2}}{4 μ_{0}} - \frac{1}{4 μ_{0}} \frac{k^{2} π^{2} B_{0}^{2}}{{(\frac{ω}{c})}^{2} - k^{2}} [{(\frac{n}{b})}^{2} + {(\frac{m}{a})}^{2}]} \int < u > \cdot da = \frac{B_{0}^{2} {abω}^{2}}{8 μ_{0} ω_{mn}^{2}} k^{2} π^{2} B_{0}^{2}}{{(\frac{ω}{c})}^{2} - k^{2}} [\begin{matrix} {(\frac{n}{b})}^{2} c o s^{2} (\frac{m π x}{a}) \\ s i n^{2} (\frac{n π y}{b}) + \\ {(\frac{m}{a})}^{2} s i n^{2} (\frac{m π x}{a}) c o s^{2} (\frac{n π y}{b}) \end{matrix}] \end{matrix}] \end{matrix}} \int < u > \cdot d a = \frac{a b}{4} {\frac{ε_{0}}{4} \frac{ω^{2} π^{2} B_{0}^{2}}{{(\frac{ω}{c})}^{2} - k^{2}} [{(\frac{n}{b})}^{2} - {(\frac{m}{a})}^{2}] + \frac{B_{0}^{2}}{4 μ_{0}} - \frac{1}{4 μ_{0}} \frac{k^{2} π^{2} B_{0}^{2}}{{(\frac{ω}{c})}^{2} - k^{2}} [{(\frac{n}{b})}^{2} + {(\frac{m}{a})}^{2}]} \int < u > \cdot d a = \frac{B_{0}^{2} a b ω^{2}}{8 μ_{0} ω_{m n}^{2}} \begin{array}{rcl} <S> & = & \frac{1}{2 μ_{0}} (B_{y}^{*} E_{x} - B_{x}^{*} E_{y}) \hat{z} \\ <S> & = & \frac{ω k π^{2} B_{0}^{2}}{{(\frac{ω}{c})}^{2} - k^{2}} [{(\frac{n}{b})}^{2} \cos^{2} (\frac{m π x}{a}) \sin^{2} (\frac{n π y}{b}) + {(\frac{m}{a})}^{2} \sin^{2} (\frac{m π x}{a}) \cos^{2} (\frac{n π y}{b}) \hat{z}] \\ \int <S> \cdot d a & = & \frac{1}{8 μ_{0}} \frac{ω k π^{2} B_{0}^{2}}{{(\frac{ω}{c})}^{2} - k^{2}} a b [{(\frac{m}{a})}^{2} + {(\frac{n}{b})}^{2}] \\ \int <S> \cdot d a & = & \frac{1}{8 μ_{0}} \frac{ω k c^{2} B_{0}^{2} a b}{(ω_{m n}^{2})} \end{array}$

Determine the group velocity:

Substitute $\int < S > \cdot d a = \frac{1}{8 μ_{0}} \frac{ω k c^{2} B_{0}^{2}}{(ω_{m n}^{2})}$ and $\int < u > \cdot d a = \frac{B_{0}^{2} a b ω^{2}}{8 μ_{0} ω_{m n}^{2}}$ in equation (1).

$V \frac{\frac{1}{8 μ_{0}} \frac{ω k c^{2} B_{0}^{2} a b}{(ω_{m n}^{3})}}{\frac{B_{0}^{2} a b ω^{2}}{8 μ_{0} ω_{m n}^{2}}} V = \frac{1}{8 μ_{0}} \frac{ω k c^{2} B_{0}^{2} a b}{(ω_{m n}^{2})} \times \frac{8 μ_{0} ω_{m n}^{2}}{B_{0}^{2} a b ω} V = \frac{k c^{2}}{ω} V = \frac{c}{ω} \sqrt{ω^{2} - ω_{m n}^{2}}$

Therefore, the velocity of energy carried by the electromagnetic wave is confirmed to be the group velocity of the wave.

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Short Answer

Step by step solution

Expression for the velocity of the energy carried by the electromagnetic wave traveling in a waveguide:

Determine the time-averaged pointing vector for the electromagnetic wave:

Determine the energy per unit volume of the electromagnetic wave:

Determine the group velocity:

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