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Chapter 20: Problem 2716

The relation between the maximum electron density \(\mathrm{N}\) max and the critical frequency \(\mathrm{f}\) for the ionosphere can be given as (A) \(\left.f_{c}=\sqrt{(9 N}_{\max }\right)\) (B) \(\mathrm{f}_{\mathrm{c}}=\sqrt{9}\left(\mathrm{~N}_{\max }\right)\) (C) \(\left.f_{c}=9 \sqrt{(N}_{\max }\right)\) (D) None of these

Short Answer

Expert verified
Based on the examination of options A, B, and C, all of them lead to a mismatch of units and don't provide a meaningful relationship between the maximum electron density (N_max) and the critical frequency (f_c) for the ionosphere. Hence, the correct answer is (D) None of these.

Step by step solution

01

Understanding the variables and their physical units

Before we examine each option, let's make sure we understand the physical meaning of the variables involved and their corresponding units. The maximum electron density, denoted by N_max, is a measure of the number of free electrons in a given volume of the ionosphere. It is expressed in SI units as m^(-3), or electrons per cubic meter. The critical frequency, denoted by f_c, represents the highest frequency of an electromagnetic wave that can be reflected by the ionosphere. It is expressed in SI units as Hz (Hertz), or cycles per second.
02

Examining Option A

First, let's consider option A: \(f_{c} = \sqrt{(9 N}_{\max})\). To check whether this equation makes physical sense, we must determine whether the square root of electron density yields a frequency. Taking the square root of N_max would result in units of m^(-3/2). Based on the units alone, this equation is not correct because a square root of electron density doesn't yield a frequency. Therefore, we can eliminate option A.
03

Examining Option B

Next, let's consider option B: \(f_{c} = \sqrt{9}(\mathrm{~N}_{\max})\). In this equation, we have N_max multiplied by a dimensionless constant, \(\sqrt{9}\), which won't result in a proper frequency dimension. Like in step 2, this equation leads to a mismatch of units and doesn't make physical sense. So, we can eliminate option B.
04

Examining Option C

Now, let's consider option C: \(f_{c} = 9 \sqrt{(N}_{\max})\). In this equation, we are multiplying the square root of N_max by a dimensionless constant, 9. As in steps 2 and 3, this equation also results in mismatched units and doesn't make physical sense. So, we can eliminate option C.
05

Conclusion

Based on our examination of options A, B, and C, all of them lead to a mismatch of units and don't provide a meaningful relationship between the maximum electron density (N_max) and the critical frequency (f_c) for the ionosphere. Hence, the correct answer is (D) None of these.

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Most popular questions from this chapter

In the following question from statement-1 (Assertion) is followed by statement-2 (Reason) Each question has the following four choice out of which only one choice is correct (a) Statement- 1 is true, statement \(-2\) is true. Statement \(-2\) is a correct explanation for statement-1. (b) Statement-1 is true, statement-2 is true, statement-2 is not a correct explanation for statement-1. (c) Statement- 1 is true, statement \(-2\) is false (d) Statement-1 is false, statement \(-2\) is true. Statement-1 :- sky wave communication is not suitable for frequencies greater than \(30 \mathrm{MHz}\). Statement \(-2:-\) High frequency signals die out before reaching the ionosphere. (A) a (B) \(b\) (C) \(c\) (D) \(\mathrm{d}\)

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