Question

On a particular day, the maximum frequency reflected from the ionosphere is \(8 \mathrm{MHz}\). On another day it was found to increase to \(9 \mathrm{MHz}\). The ratio of the maximum electron densities of ionosphere on the two days is (A) \(\sqrt{(8 / 9)}\) (B) \((8 / 9)\) (C) \((8 / 9)^{(3 / 2)}\) (D) \((8 / 9)^{2}\)

Short answer

The ratio of the maximum electron densities of ionosphere on the two days is \((\frac{8}{9})^2\).

Step-by-step solution

Critical frequency formula

The critical frequency (\(f_{c}\)), which is the highest frequency that can be reflected by the ionosphere for a given layer, is directly related to the electron density (\(N\)) of the ionosphere. This is given by the formula: \(f_{c} = 9 \sqrt{N}\) where \(f_{c}\) is in MHz and \(N\) is in electrons per cubic meter (el/m³).

Determine the electron densities

Using the given maximum frequencies, we can find the electron densities for each day using the critical frequency formula from Step 1. For day 1 (\(f_{c1} = 8\mathrm{MHz}\)): \(N_1 = \left(\frac{f_{c1}}{9}\right)^2\) For day 2 (\(f_{c2} = 9\mathrm{MHz}\)): \(N_2 = \left(\frac{f_{c2}}{9}\right)^2\)

Find the ratio of electron densities

Divide \(N_2\) by \(N_1\) to find the ratio of the maximum electron densities on the two days: \(\frac{N_2}{N_1} = \frac{(\frac{f_{c2}}{9})^2}{(\frac{f_{c1}}{9})^2} = \frac{f_{c2}^2}{f_{c1}^2}\)

Plug in given values and match the answer

Insert the given maximum frequencies into the ratio: \(\frac{N_2}{N_1} = \frac{(9 \ \mathrm{MHz})^2}{(8 \ \mathrm{MHz})^2} = \frac{81}{64} = \left(\frac{9}{8}\right)^2\) This matches option (D): \(\left(\frac{8}{9}\right)^2\). So, the ratio of the maximum electron densities of ionosphere on the two days is \((\frac{8}{9})^2\).

Key concepts

Critical Frequency

The critical frequency (\(f_c\)) is an important concept when discussing the ionosphere and its interaction with radio waves. It represents the highest frequency at which a radio wave can be reflected back to Earth by a given layer of the ionosphere. This threshold depends on the electron density (\(N\)) of that layer. The critical frequency is measured in megahertz (MHz) and is determined by the formula:
\[f_c = 9 \sqrt{N}\]- **Understanding the Formula:** - The number 9 in the equation is a constant that relates frequency to electron density. - The square root indicates that the critical frequency increases with the square root of electron density.- **Practical Implication:** Knowing the critical frequency helps in determining whether a signal will be reflected or pass through the ionosphere, crucial for radio communication and navigation systems.
It's essential to grasp that as electron density changes, so does the critical frequency, impacting radio wave propagation.

Electron Density

Electron density (\(N\)) in the ionosphere is a measure of how many free electrons are present per cubic meter. This density is crucial in influencing how radio waves interact with the ionosphere:
- **Role in Signal Reflection:** - Higher electron densities can reflect higher frequency waves back to Earth. - It's a key factor in determining the maximum usable frequency for reliable communication.- **Calculation from Critical Frequency:** - From the critical frequency (\(f_c\)), electron density can be calculated using: \[N = \left(\frac{f_c}{9}\right)^2\]- **Effect on Communication:** Variations in electron density can lead to changes in signal strength and clarity, affecting the performance of communication systems.
Understanding electron density changes helps anticipate and adjust for these variations.

Maximum Frequency

The term "maximum frequency" in the context of the ionosphere refers to the highest frequency of radio waves that can be reflected back to Earth for effective communication. This maximum frequency is closely related to both the critical frequency and the electron density:
- **Direct Relationship:** - It is theoretically the same as the critical frequency for a given layer of the ionosphere. - Changes in electron density adjust the maximum frequency accordingly. - **Measurement:** - Typically observed as the maximum frequency that doesn’t penetrate the ionosphere but reflects instead. - **Practical Use:** - In the exercise discussed, observing an increase from 8 MHz to 9 MHz suggests an increase in the maximum frequency due to a rise in electron density. This impacts how far and clear the radio signals travel.
Understanding maximum frequency helps predict communication range and quality, especially important for long-distance radio communications.