Animated Solution for Physics - Electromagnetic Waves: The correct match between the entries in table are \begin{array}{|c|c|} \hline \text{Column I (Radiation)} & \text{Column II (Wavelength)} \\ \hline \text{(A) Microwaves} & \text{(I) } 100\text{ m} \\ \text{(B) } \gamma\text{-rays} & \text{(II) } 10^{-15}\text{ m} \\ \text{(C) AM radiowaves} & \text{(III) } 10^{-10}\text{ m} \\ \text{(D) X-rays} & \text{(IV) } 10^{-3}\text{ m} \\ \hline \end{array}

Have you ever wondered how the universe communicates with us? From the warmth of the sun to the radio stations we tune into, it's all part of a grand cosmic symphony known as the electromagnetic spectrum. In this problem, we are tasked with matching different types of electromagnetic radiation with their corresponding wavelengths. It might seem like a test of memory, but it's actually a beautiful exercise in logical deduction.

The electromagnetic spectrum is a continuous range of waves that travel through space at the speed of light. These waves differ only in their frequency and wavelength. On one end of the spectrum, we have waves with incredibly long wavelengths and low frequencies, like radio waves. On the other end, we find waves with minuscule wavelengths and extremely high frequencies, such as gamma rays.

Imagine the spectrum as a grand piano. The low, rumbling bass notes represent the radio waves, while the piercing, high-pitched treble notes represent the gamma rays. Every other type of radiation—microwaves, infrared, visible light, ultraviolet, and X-rays—fits perfectly between these two extremes. What is truly mind-blowing is that the visible light we see with our eyes—all the colors of the rainbow—makes up only a tiny, almost insignificant sliver of this vast cosmic keyboard. The rest of the spectrum is entirely invisible to us, yet it interacts with our world in profound ways.

To solve this matching problem without relying purely on rote memorization, we need a master key. That key is the fundamental relationship between the energy of a photon and its wavelength, given by the famous Planck-Einstein relation:

Here, $E$ represents the energy of a single photon, $h$ is Planck's constant (a fundamental number in quantum mechanics), $c$ is the speed of light in a vacuum, and $\lambda$ is the wavelength of the wave. Because $h$ and $c$ are absolute constants of nature, we can clearly see an inverse relationship between energy and wavelength:

This simple proportionality is incredibly powerful. It tells us that the more energetic a wave is, the shorter its wavelength must be. The waves are tightly packed together, vibrating with intense frequency. Conversely, a wave with low energy will stretch out over a much longer wavelength, undulating slowly through space. When you encounter a problem asking you to match radiations with their wavelengths, this equation is your best friend. It allows you to bypass memorization and rely on pure physical intuition.

Let's look at the four types of radiation provided in Column I: Microwaves, $\gamma$-rays, AM radiowaves, and X-rays. Instead of trying to remember their exact numerical wavelengths, let's think about their physical properties, how they are generated, and what they are used for. This will allow us to easily rank them by energy.

$\gamma$-rays (Gamma Rays): These are the heavyweights of the electromagnetic spectrum. Gamma rays are produced by the most violent events in the universe, such as supernovas, black hole collisions, and the radioactive decay of atomic nuclei. Because they originate from the tightly bound nucleus of an atom, they carry an immense amount of energy. They are so energetic that they can easily pass through most materials, including human bodies, which is why they require thick lead shielding to be stopped.

X-rays: Just a step down in energy from gamma rays, we find X-rays. These are typically generated when high-speed electrons are suddenly decelerated by striking a metal target. While not as energetic as gamma rays, they still pack a significant punch. We are all familiar with X-rays from medical imaging; they have enough energy to pass right through our soft tissues but are absorbed by denser materials like our bones, creating the skeletal shadows we see on an X-ray film.

Microwaves: Moving much further down the energy scale, we encounter microwaves. These waves have significantly less energy than X-rays or visible light. They are generated by specialized vacuum tubes or solid-state devices. We use them every day in our kitchens to heat up food. They work by transferring their relatively low energy to water molecules in the food, causing them to vibrate and generate heat. Microwaves are also the backbone of modern communication, carrying signals for our mobile phones, Wi-Fi networks, and satellite transmissions.

AM Radiowaves: Finally, we reach the lowest energy waves on our list: AM (Amplitude Modulation) radiowaves. These waves are generated by oscillating electric currents in massive antennas. Because their energy is so low, they are completely harmless to us. However, their low energy corresponds to massive wavelengths, which allows them to diffract around large obstacles like mountains and buildings, and even bounce off the Earth's ionosphere. This is why an AM radio station can broadcast its signal across hundreds or even thousands of miles.

So, based on our physical understanding, our energy ranking from highest to lowest is definitively:

Because of the inverse relationship we established earlier, the order of their wavelengths will be exactly the opposite. The highest energy wave will have the shortest wavelength, and the lowest energy wave will have the longest wavelength:

Now, let's examine the specific wavelengths provided in Column II: $100\text{ m}$, $10^{-15}\text{ m}$, $10^{-10}\text{ m}$, and $10^{-3}\text{ m}$.

We simply need to arrange these numerical values from smallest to largest. It's a straightforward exercise in reading scientific notation:

By aligning our logically ordered list of radiations with this ordered list of wavelengths, the matches reveal themselves naturally and undeniably:

1. The absolute shortest wavelength, $10^{-15}\text{ m}$, must belong to the highest energy radiation on our list: $\gamma$-rays. So, (B) matches perfectly with (II). 2. The next shortest wavelength, $10^{-10}\text{ m}$, belongs to the second most energetic radiation: X-rays. So, (D) matches with (III). 3. The next value, $10^{-3}\text{ m}$ (which is one millimeter), corresponds to Microwaves. So, (A) matches with (IV). 4. Finally, the longest wavelength by far, $100\text{ m}$, belongs to the lowest energy radiation: AM radiowaves. So, (C) matches with (I).

Putting it all together, we get the final mapping: (A) $\to$ (IV), (B) $\to$ (II), (C) $\to$ (I), (D) $\to$ (III).

This perfectly aligns with option (d). By understanding the underlying physics of the electromagnetic spectrum and the fundamental relationship between energy and wavelength, we transformed a potential guessing game into a satisfying, foolproof logical deduction. Always remember, physics is not just about memorizing isolated numbers; it's about understanding the beautiful, interconnected relationships that govern our universe!