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At an instant, a plane electromagnetic wave has its magnetic field in the direction of the vector $\hat{\mathbf{i}}-\hat{\mathbf{j}}$ and its electric field is in the direction of $\hat{\mathbf{i}}+\hat{\mathbf{j}}$. The wave is travelling along which direction?
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Verified Answer
The correct answer is:
-z-direction
Direction of magnetic field,
$\hat{\mathbf{B}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}$
Direction of electric field,
$\hat{\mathbf{E}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}$
Direction of propagation of EM wave is given according to poynting vector as
$\hat{\mathbf{n}}=\hat{\mathbf{E}} \times \hat{\mathbf{B}}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}) \times(\hat{\mathbf{i}}-\hat{\mathbf{j}})$
$=\hat{\mathbf{i}} \times \hat{\mathbf{i}}+\hat{\mathbf{j}} \times \hat{\mathbf{i}}-\hat{\mathbf{i}} \times \hat{\mathbf{j}}-\hat{\mathbf{j}} \times \hat{\mathbf{j}}$
$=0+(-\hat{\mathbf{k}})-\hat{\mathbf{k}}-0=-2 \hat{\mathbf{k}}$
Hence, electromagnetic wave will travel along $-z$-direction.
$\hat{\mathbf{B}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}$
Direction of electric field,
$\hat{\mathbf{E}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}$
Direction of propagation of EM wave is given according to poynting vector as
$\hat{\mathbf{n}}=\hat{\mathbf{E}} \times \hat{\mathbf{B}}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}) \times(\hat{\mathbf{i}}-\hat{\mathbf{j}})$
$=\hat{\mathbf{i}} \times \hat{\mathbf{i}}+\hat{\mathbf{j}} \times \hat{\mathbf{i}}-\hat{\mathbf{i}} \times \hat{\mathbf{j}}-\hat{\mathbf{j}} \times \hat{\mathbf{j}}$
$=0+(-\hat{\mathbf{k}})-\hat{\mathbf{k}}-0=-2 \hat{\mathbf{k}}$
Hence, electromagnetic wave will travel along $-z$-direction.
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