The electric field of an electromagnetic wave in free space is given by $\vec{E} = 10 \cos(10^7 t + kx) \hat{j

Question

The electric field of an electromagnetic wave in free space is given by $\vec{E} = 10 \cos(10^7 t + kx) \hat{j}$ V/m, where $t$ and $x$ are in seconds and meters respectively. It can be inferred that:
(1) The wavelength $\lambda$ is $188.4 	ext{ m}$.
(2) The wave number $k$ is $0.33 	ext{ rad/m}$.
(3) The wave amplitude is $10 	ext{ V/m}$.
(4) The wave is propagating along $+x$ direction.
Which one of the following pairs of statements is correct?

Accepted Answer

1. **Analyze Amplitude:** The standard wave equation is $E = E_0 \cos(\omega t \pm kx)$. Comparing with the given equation, the amplitude $E_0 = 10 	ext{ V/m}$. Thus, statement (3) is correct.
2. **Analyze Direction:** The argument of the cosine function is $(10^7 t + kx)$. The positive sign between the time and position terms ($+$) indicates that the wave is propagating in the negative $x$-direction . Thus, statement (4) is incorrect.
3. **Analyze Wavelength:** The angular frequency is $\omega = 10^7 	ext{ rad/s}$. The speed of light in vacuum is $c = 3 	imes 10^8 	ext{ m/s}$. The wavelength $\lambda$ is given by $\lambda = \frac{2\pi c}{\omega}$.
   $$ \lambda = \frac{2\pi 	imes 3 	imes 10^8}{10^7} = 60\pi \approx 60 	imes 3.1416 = 188.5 	ext{ m} $$
   Thus, statement (1) is correct (188.4 m is a reasonable approximation).
4. **Analyze Wave Number:** The wave number $k = \frac{\omega}{c} = \frac{10^7}{3 	imes 10^8} = \frac{1}{30} \approx 0.033 	ext{ rad/m}$. Statement (2) claims it is $0.33 	ext{ rad/m}$, which is incorrect by a factor of 10.

Therefore, only statements (1) and (3) are correct.

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