Analyzing the Magnetic Field Null in Wave Reflection

The incident electric field is given as $\vec{E}_{inc}(x) = A_y \hat{a}_y e^{-j\frac{2\pi x}{3}}$ V/m, propagating in the +x direction in air. The phase constant is $\beta = \frac{2\pi}{3}$ rad/m.

An ideal electric conductor at $x=0$ requires the tangential electric field to be zero. This condition necessitates a reflected wave propagating in the -x direction, $\vec{E}_{ref}(x)$, such that $\vec{E}_{inc}(0) + \vec{E}_{ref}(0) = 0$. This leads to $\vec{E}_{ref}(x) = -A_y \hat{a}_y e^{j\beta x}$.

The corresponding magnetic fields are:

• Incident magnetic field: $\vec{H}_{inc}(x) = -\frac{A_y}{\eta_0} e^{-j\beta x} \hat{a}_z$
• Reflected magnetic field: $\vec{H}_{ref}(x) = -\frac{A_y}{\eta_0} e^{j\beta x} \hat{a}_z$

The total magnetic field in the air is the sum of the incident and reflected fields:

$\vec{H}_{total}(x) = \vec{H}_{inc}(x) + \vec{H}_{ref}(x) = \left(-\frac{A_y}{\eta_0} e^{-j\beta x} - \frac{A_y}{\eta_0} e^{j\beta x}\right) \hat{a}_z$

Using the trigonometric identity $e^{-j\theta} + e^{j\theta} = 2\cos(\theta)$, the total magnetic field simplifies to:

$\vec{H}_{total}(x) = -\frac{2 A_y}{\eta_0} \cos(\beta x) \hat{a}_z$

Nulls of the total magnetic field occur when $\vec{H}_{total}(x) = 0$, which implies $\cos(\beta x) = 0$. The general solutions for this condition are $\beta x = \frac{\pi}{2} + n\pi$, where $n$ is an integer.

Substituting the value of $\beta = \frac{2\pi}{3}$:

$\frac{2\pi}{3} x = \frac{\pi}{2} + n\pi$

Solving for the position $x$:

$x = \frac{3}{2\pi} \left(\frac{\pi}{2} + n\pi\right) = \frac{3}{4} + \frac{3n}{2}$

Due to reflection at $x=0$, the total field exists in the region $x \le 0$. We need to find the null closest to $x=0$ within this region.

• For $n = -1$: $x = \frac{3}{4} + \frac{3(-1)}{2} = \frac{3}{4} - \frac{3}{2} = -\frac{3}{4}$ m.
• For $n = -2$: $x = \frac{3}{4} + \frac{3(-2)}{2} = \frac{3}{4} - 3 = -\frac{9}{4}$ m.

The null positions in the region $x \le 0$ are ..., $-9/4, -3/4$. The first null measured from $x=0$ in this relevant region is $x = -3/4$ m.