Animated Solution for Physics - Electromagnetic Waves: A linearly polarised electromagnetic wave in vacuum is $E=3.1\cos [(1.8)z-(5.4\times 10^6)t]\hat{i}$ N/C is incident normally on a perfectly reflecting wall at $z=a$. Choose the correct option.

Imagine an electromagnetic wave surging through the vacuum of space. The mathematical signature of this wave is given to us as $E=3.1\cos [(1.8)z-(5.4\times 10^6)t]\hat{i}$ N/C. This equation is packed with physical meaning. By comparing it to the standard wave equation, $E=E_0\cos (kz-\omega t)\hat{i}$, we can immediately extract its core properties.

The amplitude of the electric field is $E_0=3.1$ N/C. The wave number, which tells us about the spatial frequency of the wave, is $k=1.8$ rad/m. The angular frequency, dictating how fast the wave oscillates in time, is $\omega =5.4\times 10^6$ rad/s. Furthermore, the negative sign in the phase term $(kz-\omega t)$ reveals that the wave is propagating in the positive $z$-direction, while the $\hat{i}$ vector indicates it is polarized along the $x$-axis.

Let's put the given options to the test. First, we evaluate the wavelength $\lambda$. The relationship between wavelength and wave number is $\lambda =\frac{2\pi }{k}$. Substituting our value, we get $\lambda =\frac{2\pi }{1.8}\approx 3.49$ m. This immediately disqualifies option (a), which claims the wavelength is $5.4$ m.

Next, we check the frequency $f$. The frequency is related to the angular frequency by $f=\frac{\omega }{2\pi }$. Plugging in our $\omega$, we find $f=\frac{5.4\times 10^6}{2\pi }\approx 8.59\times 10^5$ Hz. This proves option (b) is also incorrect.

Reflection at a Rigid Boundary

Now, the wave encounters a perfectly reflecting wall at $z=a$. A perfectly reflecting wall acts like an impenetrable barrier; it absorbs nothing and transmits nothing. Therefore, the transmitted wave is absolutely zero, rendering option (c) incorrect.

What happens to the reflected wave? When a wave bounces off a rigid boundary, its direction of propagation is completely reversed. Our incident wave was traveling in the $+z$ direction. The reflected wave must travel in the $-z$ direction. Mathematically, this means the phase term $(kz-\omega t)$ transforms into $(kz+\omega t)$.

Thus, the spatial and temporal components of the phase now share the same sign, indicating propagation in the negative direction. The equation for the reflected wave becomes $E_r=3.1\cos [(1.8)z+(5.4\times 10^6)t]\hat{i}$ N/C. This perfectly matches option (d), making it our correct answer.