Antenna Length Calculation for 900 MHz Radio Waves

This question asks us to determine the appropriate length for a transmitting antenna designed to radiate radio waves at a frequency of 900 MHz. The length of an antenna is often related to the wavelength of the radio waves it is designed to transmit or receive.

Understanding Wavelength and Frequency

Radio waves are electromagnetic waves, and their wavelength ($\lambda$) is inversely proportional to their frequency ($f$). The relationship is given by the formula:

$$ \lambda = \frac{c}{f} $$

Where:

• $c$ is the speed of light in a vacuum, approximately $3 \times 10^8$ meters per second (m/s).
• $f$ is the frequency of the radio wave in Hertz (Hz).

Calculating the Wavelength

First, we need to convert the given frequency from Megahertz (MHz) to Hertz (Hz):

Frequency ($f$) = 900 MHz = $900 \times 10^6$ Hz = $9 \times 10^8$ Hz.

Now, we can calculate the wavelength:

$$ \lambda = \frac{3 \times 10^8 \text{ m/s}}{9 \times 10^8 \text{ Hz}} $$

$$ \lambda = \frac{1}{3} \text{ meters} $$

To make comparisons easier with the options given in centimeters (cm), let's convert the wavelength to centimeters:

$$ \lambda = \frac{1}{3} \text{ m} \times 100 \text{ cm/m} $$

$$ \lambda \approx 33.33 \text{ cm} $$

Relating Wavelength to Antenna Length

Antennas are often designed to be a specific fraction of the wavelength for efficient transmission or reception. Common lengths include:

• Half-wavelength antenna ($\lambda/2$): This is a common length for dipole antennas.
• Quarter-wavelength antenna ($\lambda/4$): This is a common length for monopole antennas.

Calculating Potential Antenna Lengths

Let's calculate these common lengths based on the wavelength we found:

• Half-wavelength ($\lambda/2$): $\frac{33.33 \text{ cm}}{2} \approx 16.67 \text{ cm}$
• Quarter-wavelength ($\lambda/4$): $\frac{33.33 \text{ cm}}{4} \approx 8.33 \text{ cm}$

Comparing with Options

Now, let's compare our calculated lengths with the options provided:

The calculated quarter-wavelength ($\approx 8.33$ cm) closely matches Option 3 (8.3 cm). The calculated half-wavelength ($\approx 16.67$ cm) closely matches Option 1 (16.7 cm). Since the provided correct answer is 8.3 cm, it suggests the question is likely referring to a quarter-wavelength antenna.

Conclusion

For radiating radio waves of 900 MHz, a common antenna length is a quarter of the wavelength ($\lambda/4$). Based on the calculation, this length is approximately 8.33 cm, which aligns well with the option 8.3 cm.