Properties of Electromagnetic Wave Travelling in Vacuum

An electromagnetic wave travelling in vacuum has an electric field given by: $\mathbf{E}(\mathbf{r}, t) = E_0 \sin(3y + 4z + \omega t) \hat{i}$

Comparing this expression with the general form of a plane electromagnetic wave: $\mathbf{E}(\mathbf{r}, t) = \mathbf{E}_0 \sin(\mathbf{k} \cdot \mathbf{r} + \omega t)$

where $\mathbf{k} \cdot \mathbf{r} = k_x x + k_y y + k_z z$.

Comparing the phase terms gives: $\mathbf{k} \cdot \mathbf{r} = 3y + 4z \implies \mathbf{k} = 3\hat{j} + 4\hat{k}$

1. Direction of Propagation: Since the phase contains $+\omega t$, a wave of the form $\sin(\mathbf{k} \cdot \mathbf{r} + \omega t)$ travels in the direction opposite to $\mathbf{k}$, i.e., along $-\mathbf{k}$. The unit vector in the direction of propagation $\hat{n}$ is: $\hat{n} = -\frac{\mathbf{k}}{|\mathbf{k}|} = -\frac{3\hat{j} + 4\hat{k}}{\sqrt{3^2 + 4^2}} = -\frac{1}{5}(3\hat{j} + 4\hat{k})$ Thus, Statement (A) is correct.

2. Magnitude of the Wave Vector: $k = |\mathbf{k}| = \sqrt{3^2 + 4^2} = 5\text{ m}^{-1}$ Therefore, Statement (B) is incorrect (since it states $0.5\text{ m}^{-1}$).

3. Angular Frequency $\omega$: The speed of light in vacuum is related to the angular frequency $\omega$ and wave number $k$ by: $c = \frac{\omega}{k} \implies \omega = c \cdot k$ Given $c = 3 \times 10^8\text{ m s}^{-1}$ and $k = 5\text{ m}^{-1}$: $\omega = (3 \times 10^8) \times 5 = 15 \times 10^8\text{ rad s}^{-1} = 1.5 \times 10^9\text{ rad s}^{-1}$ Thus, Statement (C) is correct.

4. Magnetic Field Vector $\mathbf{B}$: The magnetic field associated with an electromagnetic wave is given by: $\mathbf{B} = \frac{1}{c} (\hat{n} \times \mathbf{E})$ Substituting $\hat{n} = -\frac{1}{5}(3\hat{j} + 4\hat{k})$ and $\mathbf{E} = E_0 \sin(3y + 4z + \omega t) \hat{i}$: $\mathbf{B} = \frac{1}{c} \left[ -\frac{1}{5}(3\hat{j} + 4\hat{k}) \times \hat{i} \right] E_0 \sin(3y + 4z + \omega t)$ Using the vector cross products $\hat{j} \times \hat{i} = -\hat{k}$ and $\hat{k} \times \hat{i} = \hat{j}$: $\mathbf{B} = \frac{E_0}{5c} \sin(3y + 4z + \omega t) (-4\hat{j} + 3\hat{k})$ Therefore, Statement (D) is incorrect.

Correct Answer: A, C