Question

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):

• A. \( \sqrt{14} c \)
• B. \( \sqrt{12} c \)
• C. \( \sqrt{10} c \)
• D. \( \sqrt{7} c \)

Hint:

In electromagnetic waves, the wave number \( k \) and the frequency \( \omega \) are related by the equation \( k = \frac{\omega}{c} \), where \( c \) is the speed of light.

Answer 1

The Correct Option is A

Step 1: Understanding the wave equation.
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), \( \beta = \sqrt{14} c \).