Question

A uniform plane wave with electric field: \[ \vec{E}(x) = A_y \hat{a}_y e^{-\frac{j2\pi x}{3}} \, {V/m}, \] is traveling in the air (relative permittivity, \(\epsilon_r = 1\), and relative permeability, \(\mu_r = 1\)) in the \(+x\) direction. It is incident normally on an ideal electric conductor (conductivity, \(\sigma = \infty\)) at \(x = 0\). The position of the first null of the total magnetic field in the air (measured from \(x = 0\), in meters) is \(\_\_\_\_\).

• A. \(-\frac{3}{4}\)
• B. \(-\frac{3}{2}\)
• C. \(-6\)
• D. \(-3\)

Hint:

For standing wave problems, identify the wavelength using \(\lambda = \frac{2\pi}{\beta}\), and use the null condition to find specific positions like \(-\frac{\lambda}{4}\).

Answer 1

The Correct Option is A

Step 1: Understand the standing wave condition.
A standing wave is formed due to the reflection of the magnetic field at the boundary \(x = 0\). The null points of the magnetic field occur at positions where: \[ x = -\frac{\lambda}{4}, -\frac{3\lambda}{4}, \dots \] Step 2: Calculate the wavelength.
The wavelength \(\lambda\) is related to the propagation constant \(\beta\) by: \[ \beta = \frac{2\pi}{\lambda} \quad \Rightarrow \quad \lambda = \frac{2\pi}{\beta}. \] Substitute \(\beta = \frac{2\pi}{3}\): \[ \lambda = 3 \, {m}. \] Step 3: Determine the position of the first null.
The first null occurs at: \[ x = -\frac{\lambda}{4} = -\frac{3}{4} \, {m}. \] Final Answer: \[ \boxed{{(1) } -\frac{3}{4}} \]