For sky waves, following statements are given : (A) n > 1, this shows 81N/f2

 

56) For sky waves, following statements are given :

• (A) n > 1, this shows
  81N/f²
  positive
• (B) n < 1, shows
  81N/f²
  Negative
• (C) n < 1 shows
  81N/f²
  < 0
• (D) v_g · v_p = c²
• (E) n = 0 shows 
  81N/f²
  = 1, f = f_c

Choose the correct answer from the options given below :

• (1) (A), (D) only
• (2) (B), (D) only
• (3) (B), (C), (D), (E) only
• (4) (A), (D), (E) only

Answer: option C

To find the solution for question 56, we need to analyze the properties of sky waves propagating through the ionosphere.

Analysis of the Statements:

• Refractive Index (

  $n$

  ) and Curvature:
  The refractive index of an ionospheric layer is given by: 

  $n=\sqrt{1-\frac{81N}{f^2}}$

  , where 

  $N$

   is electron density and 

  $f$

   is frequency.

  • Since 

    $\frac{81N}{f^2}$

     is always positive,  is always less than 1 (

    $n<1$

    ). Statement (A) is incorrect.

  • As a sky wave travels upward into the ionosphere, the electron density (

    $N$

    ) increases, which causes the refractive index (

    $n$

    ) to decrease. Mathematically, 

    $\frac{dn}{dz}$

     is negative.

  • The path of a wave is related to the derivative of 

    $n$

    . For a wave to bend back toward Earth, its second derivative (

    $\frac{d^{2}z}{d{x}^2}$

    ) must be negative. Since 

    $n<1$

     in the ionosphere and it bends downward, statements (B) and (C) are both correct. They represent the same physical condition (negative curvature).

• Velocity Relation (Statement D):
  In a dispersive medium like the ionosphere, the relationship between phase velocity (

  $v_p=\frac{c}{n}$

  ) and group velocity (

  $v_g=c\cdot n$

  ) is:
  

  $$v_g\cdot v_p=(c\cdot n)\cdot \left(\frac{c}{n}\right)=c^2$$

  
  Thus, statement (D) is correct.

• Critical Frequency (Statement E):
  The critical frequency (

  $f_c$

  ) is the highest frequency returned from a layer at vertical incidence. At this point, the refractive index 

  $n$

   becomes zero at the peak of the layer.
  

  $$n=0⟹1-\frac{81N}{f^2}=0⟹\frac{81N}{f^2}=1\text{ and }f=f_c$$

  
  Thus, statement (E) is correct.

Conclusion:

Statements (B), (C), (D), and (E) are all correct based on ionospheric propagation theory.

Previous yr Question papers with Full Explanations →

Electronics and Communiaction Study Materials →