This question has Statement-1 and Statement-2. Of the four choices given after the Statements, choose the one that best describes the two Statements.
Statement-1: Short wave transmission is achieved due to the total internal reflection of the e-m wave from an appropriate height in the ionosphere.
Statement-2: Refractive index of a plasma is independent of the frequency of e-m waves.

Let us understand and evaluate both statements step by step.

First, consider Statement-1: "Short wave transmission is achieved due to the total internal reflection of the e-m wave from an appropriate height in the ionosphere." Short wave transmission refers to the propagation of electromagnetic waves in the high-frequency (HF) range, typically between 3 MHz and 30 MHz. These waves are used for long-distance communication because they can be reflected back to Earth by the ionosphere. The ionosphere is a layer of the Earth's upper atmosphere that is ionized by solar radiation, making it a plasma. As the electromagnetic wave travels upwards into the ionosphere, it encounters regions with increasing electron density. The refractive index of the ionosphere decreases with height due to this increasing electron density. According to the principles of wave propagation, when a wave moves from a region of higher refractive index to lower refractive index, it can undergo reflection if the conditions are right. In the ionosphere, this gradual change in refractive index causes the wave to bend continuously and eventually reflect back towards the Earth, similar to total internal reflection. This process allows short waves to travel long distances by bouncing between the ionosphere and the Earth's surface. Therefore, Statement-1 is correct.

Now, consider Statement-2: "Refractive index of a plasma is independent of the frequency of e-m waves." A plasma, like the ionosphere, has a refractive index that depends on the frequency of the electromagnetic wave passing through it. The refractive index $$n$$ for a plasma is given by the formula:

$$n = \sqrt{1 - \frac{\omega_p^2}{\omega^2}}$$

Here, $$\omega_p$$ is the plasma frequency, which depends on the electron density $$n_e$$, the electron charge $$e$$, the electron mass $$m_e$$, and the permittivity of free space $$\epsilon_0$$. The plasma frequency is expressed as:

$$\omega_p = \sqrt{\frac{n_e e^2}{m_e \epsilon_0}}$$

In the formula for $$n$$, $$\omega$$ is the angular frequency of the electromagnetic wave, related to its frequency $$f$$ by $$\omega = 2\pi f$$. Clearly, the refractive index $$n$$ depends on $$\omega$$, the frequency of the wave. For example, if the wave frequency $$\omega$$ is less than $$\omega_p$$, $$n$$ becomes imaginary, indicating that the wave cannot propagate and is reflected. If $$\omega$$ is greater than $$\omega_p$$, $$n$$ is real and less than 1, allowing propagation. Since $$n$$ changes with $$\omega$$, Statement-2, which claims independence from frequency, is false.

Moreover, Statement-2 cannot explain Statement-1 because if the refractive index were independent of frequency, all electromagnetic waves would behave similarly in the ionosphere. However, in reality, higher-frequency waves (like VHF or UHF) may penetrate the ionosphere without reflection, while lower-frequency short waves are reflected. The frequency dependence is crucial for the reflection mechanism described in Statement-1.

Hence, Statement-1 is true, but Statement-2 is false. Therefore, the correct choice is Option A.