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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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Answer» 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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