Given E= 10e^-j(4x - kt) - V/m in free space. (a) Write all the four Maxwell’s equations in free space. (b) Find ∆ × E. (c) Find H.

Question

Given [latex]E=10 e^{-j(4 x-k t)}[/latex] - V/m in free space.

(a) Write all the four Maxwell’s equations in free space.

(b) Find ∆ × E.

(c) Find H.

Accepted Answer

Given, [latex]E=10 e^{-4 j(4 x-k t)} a_{y} V / m[/latex] represents uniform plane wave travelling in x direction with velocity v – k/4 and [latex]E_{x}=0, E_{2}=0[/latex]

So,

[latex]\frac{\partial}{\partial y} E_{y}=-j 40 e^{-j(4 x-k t)}[/latex]

[latex] abla imes \vec{E}=\left|\begin{array}{ccc}a_{x} & a_{y} & a_{z} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \E_{x} & E_{y} & E_{z}\end{array} ight|=\left|\begin{array}{ccc}a_{x} & a_{y} & a_{z} \\frac{\partial}{\partial y} & 0 & 0 \0 & E_{y} & 0\end{array} ight|[/latex]

[latex]\begin{aligned} abla imes \vec{E} &=\vec{a}_{x}(0)-a_{y}(0)+a_{2} \cdot \frac{\partial}{\partial y} E_{y} \&=j 40 e-^{j(4 x-k t)}=40 e^{j(4 x-k t+90)}\end{aligned}[/latex]

(c) For uniform plane wave, [latex]\eta=\frac{E}{H}, \quad E=E_{y} \hat{a}_{y}[/latex] Since, wave is travelling in x direction so, the direction of [latex]\vec{H}[/latex] is in z-direction.

[latex]\begin{aligned}&\hat{y}+\hat{z}=\hat{x} \&\eta=\eta_{0}=120 \pi=\frac{E_{y}}{A z} \&H_{2}=\frac{E_{y}}{120 \pi}=\frac{10}{120 \pi} e^{-j(4 x-k t)} \&\vec{H}=H_{2} \vec{a}_{z}=\frac{1}{12 \pi} e^{-j(4 x-k t)} \vec{a}_{2}\end{aligned}[/latex]

Question 1.5.5: Given E= 10e^-j(4x - kt) - V/m in free space. (a) Write all ...

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