Question

The maximum electron density of a layer of the ionosphere is \(9 \times 10^{12} \mathrm{~m}^{-3}\). The critical frequency of this layer is (A) \(9 \mathrm{MHz}\) (B) \(6 \mathrm{MHz}\) (C) \(27 \mathrm{MHz}\) (D) \(3 \mathrm{GHz}\)

Short answer

The critical frequency, \(f_c\), can be calculated using the formula \(f_c = \dfrac{1}{2\pi} \sqrt{\dfrac{e^2N}{\varepsilon_0 m_e}}\), where the maximum electron density, N, is given as \(9 \times 10^{12} \mathrm{~m}^{-3}\). Plugging in the values and constants into the formula, we find that the critical frequency \(f_c \approx 9 \mathrm{MHz}\). Thus, the correct answer is (A) \(9 \mathrm{MHz}\).

Step-by-step solution

Plug in the values

Plug the given values and constants into the formula: \(f_c = \dfrac{1}{2\pi} \sqrt{\dfrac{(1.6 \times 10^{-19})^2(9 \times 10^{12})}{(8.85 \times 10^{-12})(9.11 \times 10^{-31})}}\)

Calculate the critical frequency

Now, perform the calculation to find the critical frequency: \(f_c = \dfrac{1}{2\pi} \sqrt{\dfrac{(2.56 \times 10^{-38})(9 \times 10^{12})}{(8.85 \times 10^{-12})(9.11 \times 10^{-31})}}\) \(f_c = \dfrac{1}{2\pi} \sqrt{\dfrac{2.304 \times 10^{-25}}{8.06 \times 10^{-42}}}\) \(f_c = \dfrac{1}{2\pi} \sqrt{2.857 \times 10^{16}}\) \(f_c \approx 9 \times 10^{6}\) Thus, the critical frequency \(f_c \approx 9 \mathrm{MHz}\). The correct answer is (A) \(9 \mathrm{MHz}\).

Key concepts

Electron Density

In the context of ionospheric physics, electron density is a crucial parameter that signifies how many free electrons are present in a specific volume of the ionosphere. It is often symbolized by the letter \( N \) and measured in electrons per cubic meter \( \text{e}^-\, \text{m}^{-3} \). The electron density can vary greatly depending on several factors like time of day, solar activity, and geographic location.
The ionosphere is divided into layers, each varying in electron density. The electron density affects how radio waves travel through or bounce off the ionosphere. When the electron density is high, the ionosphere can reflect radio waves more effectively, which impacts communication and navigation systems.

• Higher electron density means more free electrons that can affect wave propagation.
• The layers of the ionosphere are not static; their electron density changes rapidly, requiring constant monitoring for applications like GPS and telecommunication.

Ionospheric Physics

Ionospheric physics is the study of the ionized part of Earth's upper atmosphere, known as the ionosphere. This layer ranges from 50 km to several hundred kilometers above Earth and contains charged particles that affect atmospheric phenomena.
The ionosphere is influenced by solar radiation, which ionizes the gases, creating free electrons and ions. This process is more significant during daylight hours and at specific times of the year.

• The ionosphere enables radio wave reflection, allowing for longer-distance radio communication.
• It plays a crucial role in phenomena such as auroras and influences satellite communications and GPS accuracy.
• Scientists use instruments like ionosondes and radars to study its properties.

Understanding the ionosphere is vital for predicting and enhancing the performance of technology reliant on radio wave propagation.

Maxwell's Equations

Maxwell's equations form the foundation of classical electromagnetism, governing the behavior of electric and magnetic fields. They consist of four equations that describe how electric charges produce electric fields, how electric currents create magnetic fields, and how they interact.
In the ionosphere, Maxwell's equations help explain the effects of electromagnetic waves as they interact with charged particles, influencing how radio waves are transmitted and received.

• Gauss's law describes how electric charges generate electric fields.
• Faraday's law explains how changing magnetic fields can induce electric currents.
• These equations also predict how electromagnetic waves can propagate through a charged medium like the ionosphere.

Understanding these equations is crucial for applications involving radio frequencies and other aspects of ionospheric science.

Plasma Frequency

Plasma frequency, or natural frequency of oscillation, is a fundamental property in plasma physics, which is essential for understanding ionospheric behavior. It's defined as the frequency at which electrons in a plasma oscillate when subjected to an external electromagnetic field.
In the ionosphere, the critical frequency of a layer is closely related to the plasma frequency and is determined by the electron density. It's calculated using the formula for electron plasma frequency:
\[\omega_p = \sqrt{\frac{N e^2}{\varepsilon_0 m_e}}\]
where \( \omega_p \) is the plasma frequency, \( N \) is electron density, \( e \) is the charge of an electron, \( \varepsilon_0 \) the permittivity of free space, and \( m_e \) the electron mass.

• High plasma frequency implies high electron density, reflecting the ionosphere's effectiveness in reflecting radio waves.
• The relationship between plasma frequency and critical frequency is crucial for communication and radar systems.

Thus, knowing the plasma frequency helps in predicting signal propagation and designing communication systems.