Question

For transmitting a signal of frequency 1000 kHz, the minimum length of the antenna is

• A. 30 m
• B. 50 m
• C. 75 m
• D. 1500 m

Hint:

- Relationship between speed (\(c\)), frequency (\(f\)), and wavelength (\(\lambda\)) of an electromagnetic wave: \( c = f\lambda \). - Speed of light/EM waves in vacuum/air: \( c \approx 3 \times 10^8 \, \text{m/s} \). - For efficient radiation, antenna length \(L\) is typically a fraction of the wavelength, such as: - Quarter-wave monopole: \( L = \lambda/4 \) - Half-wave dipole: Total length \( L = \lambda/2 \) (each arm \( \lambda/4 \)) - "Minimum length" often refers to \( \lambda/4 \).

Answer 1

The Correct Option is C

Frequency of the signal \( f = 1000 \, \text{kHz} = 1000 \times 10^3 \, \text{Hz} = 10^6 \, \text{Hz} \).
For efficient transmission and reception, the length of the antenna (L) is typically related to the wavelength \( (\lambda) \) of the signal.
A common minimum length for an antenna (like a quarter-wave monopole or a half-wave dipole) is \( L = \lambda/4 \) or \( L = \lambda/2 \).
The question asks for "minimum length", often referring to \( \lambda/4 \).
First, calculate the wavelength \( \lambda \).
The speed of electromagnetic waves (radio waves) is the speed of light \( c = 3 \times 10^8 \, \text{m/s} \).
The relation is \( c = f\lambda \).
\[ \lambda = \frac{c}{f} = \frac{3 \times 10^8 \, \text{m/s}}{10^6 \, \text{Hz}} = 3 \times 10^{8-6} \, \text{m} = 3 \times 10^2 \, \text{m} = 300 \, \text{m} \] For a quarter-wave monopole antenna, the minimum length is \( L = \frac{\lambda}{4} \).
\[ L = \frac{300 \, \text{m}}{4} = 75 \, \text{m} \] If it were a half-wave dipole, \( L = \lambda/2 = 300/2 = 150 \) m.
Since 75 m is an option and often considered a practical "minimum" for efficient radiation (e.
g.
, for a Marconi antenna or a simple whip antenna grounded), this is the likely intended answer.
This matches option (3).