Question

Consider an optical communication system operating at \(\lambda=800 \mathrm{~nm}\). Suppose only \(1 \%\) of the optical source frequency is the available channel bandwidth for optical communication. No. of channels that can be accommodated for transmitting video \(\mathrm{T}\). \(\mathrm{V}\) signals requiring an approximate band width of \(4.5 \mathrm{MHz}\) is (A) \(3.8 \times 10^{+14}\) (B) \(3.8 \times 10^{12}\) (C) \(8.4 \times 10^{5}\) (D) \(1.71 \times 10^{7}\)

Short answer

The number of channels that can be accommodated for transmitting TV signals is approximately \(8.4 \times 10^5\) (Option C).

Step-by-step solution

Calculate the frequency of the optical communication system

We are given the wavelength (\(\lambda\)) of the optical communication system, and we need to find the frequency (\(f\)). We'll use the relation between wavelength, frequency, and the speed of light (c): \[ f = \frac{c}{\lambda} \] Given: \(\lambda = 800\) nm and the speed of light \(c = 3 \times 10^8\) m/s. First, convert the wavelength from nm to meters: \[ \lambda = 800\, nm = 800 \times 10^{-9}\, m \] Now, we can calculate the frequency of the optical communication system: \[ f = \frac{3 \times 10^8\, m/s}{800 \times 10^{-9}\, m} = 3.75 \times 10^{14}\, Hz \]

Calculate the available bandwidth

Now that we have the frequency of the optical communication system, we need to find the available bandwidth for communication. Since only 1% of the optical source frequency is available for communication, we have: \[ \text{Available bandwidth} = 0.01 \times f = 0.01 \times 3.75 \times 10^{14}\, Hz = 3.75 \times 10^{12}\, Hz \]

Calculate the number of channels

Now that we have the available bandwidth, we divide it by the required bandwidth for TV signals to find the number of channels that can be accommodated: Given: Required bandwidth for TV signals = \(4.5\) MHz First, convert the bandwidth from MHz to Hz: \[ \text{Required bandwidth} = 4.5\, MHz = 4.5 \times 10^6\, Hz \] Now, divide the available bandwidth by the required bandwidth: \[ \text{Number of channels} = \frac{\text{Available bandwidth}}{\text{Required bandwidth}} = \frac{3.75 \times 10^{12}\, Hz}{4.5 \times 10^6\, Hz} = 8.33 \times 10^5 \] Thus, the number of channels that can be accommodated for transmitting TV signals is approximately \(8.33 \times 10^5\). Among the given options, this is closest to \(8.4 \times 10^5\) (Option C). Therefore, the correct answer is (C) \(8.4 \times 10^{5}\).

Key concepts

Wavelength to Frequency Conversion

To understand how optical communication systems work, one must grasp the relationship between wavelength and frequency. These concepts are interconnected through the speed of light, particularly in systems operating at optical wavelengths. The frequency (\(f\)) of a wave is inversely proportional to its wavelength (\(\lambda\)). This relationship is expressed by the equation:
\[f = \frac{c}{\lambda}\]where \(c\) is the speed of light, approximately \(3 \times 10^8\) meters per second. To convert a wavelength given in nanometers (nm) to meters (m), multiply it by \(10^{-9}\). For example, an optical communication system with a wavelength of \(800\) nm can be converted to meters as follows:

• \(\lambda = 800 \times 10^{-9}\) m

Using this conversion, the frequency can be calculated, shedding light on how such systems transmit signals across vast distances.

Channel Bandwidth Calculation

In optical communication, the available bandwidth plays a critical role in determining the system's capacity. Bandwidth refers to the range of frequencies that can be used for communication. It's crucial as it dictates how much data can be transmitted over a channel.When only a certain percentage of the optical source frequency is available, calculating the channel bandwidth becomes necessary. For instance, if only 1% of the optical source frequency is available for communication, the bandwidth can be calculated using:

• \(\text{Available Bandwidth} = 0.01 \times f\)

where \(f\) represents the frequency obtained from the optical source. By identifying the available bandwidth, one can determine how many channels fit within the system, crucial for applications like TV broadcasting.

Optical Source Frequency

The optical source frequency is central to determining the nature and efficiency of an optical communication system. This frequency indicates how many times a wave oscillates in one second and is directly linked to the speed at which data travels through the system. By determining the frequency, you get a picture of the available spectrum for transmission. The higher the frequency, the greater the potential bandwidth, and thus more data can be sent simultaneously. Calculating this frequency involves converting the given wavelength of the optical source into frequency using the speed of light formula. In practical terms, once you have the optical source frequency, calculating things like the number of TV channels you can transmit becomes straightforward. First, establish the frequency using the original wavelength, and from there, assess how much bandwidth can be allocated for specific services.