12 In a plane electromagnetic wave, the directions of electric field and magnetic field are represented by \( \hat{ k } \) and \( 2 \hat{ i }-2 \hat{ j } \), respectively. What is the unit vector along direction of propagation of the wave? (a) \( \frac{1}{\sqrt{2}}(\hat{ i }+\hat{ j }) \) (b) \( \frac{1}{\sqrt{2}}(\hat{ j }+\hat{ k }) \) (c) \( \frac{1}{\sqrt{5}}(\hat{ i }+2 \hat{ j }) \) (d) \( \frac{1}{\sqrt{5}}(2 \hat{ i }+\hat{ j }) \)

Monjur Alahi

12 In a plane electromagnetic wave, the directions of electric field and magnetic field are represented by \( \hat{ k } \) and \( 2 \hat{ i }-2 \hat{ j } \), respectively. What is the unit vector along direction of propagation of the wave? (a) \( \frac{1}{\sqrt{2}}(\hat{ i }+\hat{ j }) \) (b) \( \frac{1}{\sqrt{2}}(\hat{ j }+\hat{ k }) \) (c) \( \frac{1}{\sqrt{5}}(\hat{ i }+2 \hat{ j }) \) (d) \( \frac{1}{\sqrt{5}}(2 \hat{ i }+\hat{ j }) \)


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Salim Ahmed Kawsar

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Correct answer is (a)

We know that propagation wave vector

∵ \(\vec{E} = \hat{k}\)

\(\vec{B} = 2\hat{i} - 2\hat{j}\)

\(\vec{C} = \vec{E} \times \vec{B}\)

\(\vec{C} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 0 & 1 \\ 2 & -2 & 0 \end{vmatrix}\)

\(\vec{C} = \hat{i}(0 + 2) - \hat{j}(0 - 2) + 0\,\hat{k}\)

\(\vec{C} = 2\hat{i} + 2\hat{j}\)

unit vector of propagation wave = \(\frac{\vec{C}}{|\vec{C}|}\)

\(=\frac{2\hat{i} + 2\hat{j}}{\sqrt{4 + 4}}\)

\(=\frac{2\hat{i} + 2\hat{j}}{\sqrt{8}}\)

\(\Rightarrow \frac{1}{\sqrt{2}}(\hat{i} + \hat{j})\)

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