Question

An electromagnetic wave travelling in vacuum has its electric field component, $E=15\sin(1.57y+5.4t)\hat{j}$. The wavelength of the wave is:

• A. 4.0 m
• B. 3.0 m
• C. 2.5 m
• D. 2.0 m
• E. 1.0 m

Hint:

Recognize $1.57$ as $\pi/2$ to simplify calculations: $\lambda = 2\pi / (\pi/2) = 4$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires extracting information about an electromagnetic wave, specifically its wavelength, from its mathematical representation.
Step 2: Key Formula or Approach:
The standard form of a sinusoidal plane wave travelling along the y-axis is: \[ E(y, t) = E_0 \sin(ky \pm \omega t + \phi) \] where: - \( E_0 \) is the amplitude. - k is the angular wave number. - \( \omega \) is the angular frequency. The angular wave number k is related to the wavelength \( \lambda \) by the formula: \[ k = \frac{2\pi}{\lambda} \] We can rearrange this to find the wavelength: \[ \lambda = \frac{2\pi}{k} \] Step 3: Detailed Explanation:
The given equation for the electric field is: \[ E = 15 \sin(1.57y + 5.4t) \] By comparing this to the standard form, we can identify the angular wave number k, which is the coefficient of the spatial variable (y). \[ k = 1.57 \text{ rad/m} \] The value 1.57 is a common approximation for \( \pi/2 \). Let's check if this assumption simplifies the problem and matches the options. Let's assume \( k = 1.57 \approx \frac{\pi}{2} \). Now, use the formula to find the wavelength \( \lambda \): \[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{\pi/2} = 2\pi \times \frac{2}{\pi} = 4 \] The units will be in meters, as k is in rad/m. So, the wavelength is 4.0 m. Step 4: Final Answer:
The wavelength of the wave is 4.0 m.