Question 8

1. The phase of the wave is given by $\phi = 3y + 4z + \omega t$.

Comparing this with the general form $(\vec{k} \cdot \vec{r} + \omega t)$, we identify the wave vector $\vec{k} = 3\hat{j} + 4\hat{k}$.

Since the signs of the spatial term and the temporal term are the same, the wave propagates in the $-\vec{k}$ direction. The direction of propagation is $\hat{n} = -\frac{\vec{k}}{|\vec{k}|} = -\frac{3\hat{j} + 4\hat{k}}{\sqrt{3^2 + 4^2}} = -\frac{1}{5}(3\hat{j} + 4\hat{k})$. Thus, option (A) is correct.

2. The magnitude of the wave vector is $k = |\vec{k}| = \sqrt{3^2 + 4^2} = 5 \text{ m}^{-1}$. Thus, option (B) is incorrect.

3. The angular frequency $\omega$ is related to $k$ and $c$ by $\omega = ck = (3 \times 10^8 \text{ ms}^{-1})(5 \text{ m}^{-1}) = 1.5 \times 10^9 \text{ rad s}^{-1}$. Thus, option (C) is correct.

4. The magnetic field $\vec{B}$ is given by $\vec{B} = \frac{\vec{k} \times \vec{E}}{\omega}$. Substituting the values:

$\vec{B} = \frac{(3\hat{j} + 4\hat{k}) \times (E_0 \sin \phi \hat{i})}{\omega} = \frac{E_0 \sin \phi}{\omega} [3(\hat{j} \times \hat{i}) + 4(\hat{k} \times \hat{i})] = \frac{E_0 \sin \phi}{\omega} [4\hat{j} - 3\hat{k}]$

Since $\omega = 5c$, $\vec{B} = \frac{E_0}{5c} \sin(3y + 4z + \omega t)(4\hat{j} - 3\hat{k})$. Option (D) is incorrect because it lacks the factor of $1/5$.