Question

The electric field associated with an electromagnetic wave travelling in vacuum is given by \[ \vec E = E_0\sin(3y+4z+\omega t)\,\hat i \] where \(\omega\) is the angular frequency. All quantities are in SI units. The correct statement(s) about this wave is/are: \[ [\text{Given: speed of light in vacuum }c=3\times10^8\ \text{m s}^{-1}] \]

• A. The wave is travelling in \(-\dfrac15(3\hat j+4\hat k)\) direction.
• B. The magnitude of wave vector is \(5\ \mathrm{m^{-1}}\)
• C. The value of \(\omega\) is \(1.5\times10^9\ \mathrm{rad\,s^{-1}}\)
• D. The magnetic field associated with this wave is given by \[ \vec B= \frac{E_0}{c} \sin(3y+4z+\omega t)\, (4\hat j-3\hat k) \]

Hint:

For electromagnetic waves: \[ \omega=ck \] and: \[ \vec E\times\vec B \] gives the direction of propagation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
An electromagnetic wave in vacuum is a transverse wave where the electric field \( \vec{E} \), magnetic field \( \vec{B} \), and propagation direction \( \hat{n} \) are mutually perpendicular. The phase of the wave is given by \( (\vec{k} \cdot \vec{r} + \omega t) \). The wave vector \( \vec{k} \) points in the direction of propagation if the time term has a negative sign (\( \vec{k} \cdot \vec{r} - \omega t \)). If both have the same sign, the wave travels in the \( -\hat{k} \) direction.
Step 2: Key Formula or Approach:
1. Wave vector: \( \vec{k} \) is the vector of spatial coefficients.
2. Direction of propagation: \( \hat{n} = -\vec{k}/|\vec{k}| \) (when signs are same).
3. Relation between \( \omega, c, k \): \( \omega = c |\vec{k}| \).
4. Magnetic field relation: \( \vec{B} = \frac{1}{c} (\hat{n} \times \vec{E}) \).
Step 3: Detailed Explanation:
From the expression \( 3y + 4z + \omega t \), the spatial part is \( \vec{k} \cdot \vec{r} = 0x + 3y + 4z \).
Thus, \( \vec{k} = 3\hat{j} + 4\hat{k} \).
The magnitude is \( k = \sqrt{3^2 + 4^2} = 5 \text{ m}^{-1} \). (Statement B is incorrect).
Direction of propagation: Since the signs of spatial and time terms are both positive, the wave travels in the \( -\vec{k} \) direction.
Unit vector \( \hat{n} = -\frac{1}{5}(3\hat{j} + 4\hat{k}) \). (Statement A is correct).
Angular frequency:
\[ \omega = c k = (3 \times 10^8 \text{ ms}^{-1})(5 \text{ m}^{-1}) = 15 \times 10^8 = 1.5 \times 10^9 \text{ rad s}^{-1} \]
(Statement C is correct).
Magnetic Field:
\[ \vec{B} = \frac{1}{c} (\hat{n} \times \vec{E}) = \frac{1}{c} \left[ -\frac{1}{5}(3\hat{j} + 4\hat{k}) \times E_0 \sin(\dots) \hat{i} \right] \]
Using cross products: \( \hat{j} \times \hat{i} = -\hat{k} \) and \( \hat{k} \times \hat{i} = \hat{j} \).
\[ \vec{B} = -\frac{E_0 \sin(\dots)}{5c} [3(-\hat{k}) + 4(\hat{j})] = \frac{E_0 \sin(\dots)}{5c} (3\hat{k} - 4\hat{j}) \]
Statement (D) gives \( \frac{E_0}{c} \dots (4\hat{j} - 3\hat{k}) \), which is in the opposite direction (and missing the factor of \( 1/5 \)). Thus, (D) is incorrect.
Step 4: Final Answer:
The wave propagation vector is determined by the coefficients of spatial variables. The direction is the negative unit vector because of identical signs in the phase. The frequency relates to the wave vector magnitude via the speed of light.