Question

The electric field of an electromagnetic wave in free space is given by \( \vec{E} = 5\sin\left(\tfrac{2\pi}{3}z - \omega t\right)\hat{y} \) V/m. Which of the following statements is correct?

• A. The wave propagates along $\hat{y}$
• B. The wave vector is given $\vec{k}=\tfrac{2\pi}{3}\hat{z}$
• C. The wavelength of the electromagnetic wave is $\frac{1}{3}$ m
• D. The corresponding magnetic field is $\vec{B}=\tfrac{5}{c}\cos(\tfrac{2\pi}{3}z-\omega t)\hat{x}$ T
• E. The frequency of the wave is approximately $10^6$ Hz

Hint:

The wave propagates in the direction of the variable inside the sine function (here, $z$). The unit vector $\hat{y}$ indicates the direction of the electric field oscillation, not the path of the wave.

Answer 1

The Correct Option is B

Concept:
The general mathematical form for a plane electromagnetic wave traveling in the positive $z$-direction is: \[ \vec{E}(z, t) = E_0 \sin(kz - \omega t)\hat{n} \] where $k$ is the wave number, $\omega$ is the angular frequency, and $\hat{n}$ is the polarization direction.

Step 1: Extract the wave vector ($k$).
By comparing the given equation $\vec{E}=5\sin(\tfrac{2\pi}{3}z-\omega t)\hat{y}$ to the standard form:
• The term inside the sine function multiplying $z$ is the magnitude of the wave vector ($k$).
• $k = \frac{2\pi}{3}$ rad/m. Since the wave varies with $z$, the wave propagates along the $z$-axis. Thus, the wave vector is: \[ \vec{k} = k\hat{z} = \frac{2\pi}{3}\hat{z} \] This confirms that statement (B) is mathematically correct.

Step 2: Evaluate wavelength ($\lambda$).
The relationship between the wave number and wavelength is defined as: \[ k = \frac{2\pi}{\lambda} \implies \lambda = \frac{2\pi}{k} \] Substituting the value of $k$: \[ \lambda = \frac{2\pi}{2\pi/3} = 3 \text{ m} \] Statement (C) claims $\lambda = 1/3$ m, which is incorrect.

Step 3: Analyze propagation and frequency.

• Propagation: The wave oscillates in the $y$-direction ($\hat{y}$) but propagates in the $z$-direction. Thus, (A) is incorrect.
• Frequency: $\omega = ck = (3 \times 10^8) \times \frac{2\pi}{3} = 2\pi \times 10^8$. Since $\omega = 2\pi f$, the frequency $f = 10^8$ Hz. Statement (E) is incorrect.