The electric field of an electromagnetic wave is given by \( \mathbf{E} = (2\hat{k} - 3\hat{j}) \times 10^{-3} \sin \left( 10^7 (x + 2y + 3z - \beta t) \right) \). The value of \( \beta \) is (c is the speed of light):
Show Hint
- \( \sqrt{14} c \)
- \( \sqrt{12} c \)
- \( \sqrt{10} c \)
- \( \sqrt{7} c \)
The Correct Option is A
Solution and Explanation
The given electric field corresponds to an electromagnetic wave with the general form \( \mathbf{E} = E_0 \sin(kx + ky + kz - \beta t) \). Comparing the arguments of the sine functions, we see that \( k = 10^7 \) and \( \beta = c \). By using the relation \( k = \frac{\omega}{c} \), we can solve for \( \beta \). After simplifying, we get \( \beta = \sqrt{14} c \).
Step 2: Conclusion.
Thus, the correct answer is option (A), \( \beta = \sqrt{14} c \).
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