Question

If the band width is \(1.6 \times 10^{12} \mathrm{~Hz}\), how many communication channels can be obtained with \(16 \mathrm{KHz}\) band width radio signal ? (A) \(10^{15}\) (B) \(10^{11}\) (C) \(10^{10}\) (D) \(10^{8}\)

Short answer

The number of communication channels that can be obtained with the given bandwidth is \(10^8\). Hence, the correct answer is option (D).

Step-by-step solution

Convert Bandwidth per Channel to Hz

First, we will convert the \(16 \mathrm{kHz}\) bandwidth per communication channel into Hz (Hertz) units: \(16 \mathrm{kHz} = 16 \times 10^3 \mathrm{Hz}\)

Calculate Number of Communication Channels

Now we will divide the total bandwidth by the bandwidth per communication channel to find the number of communication channels: Number of channels = Total bandwidth / Bandwidth per channel Number of channels = \( (1.6 \times 10^{12} \mathrm{Hz}) / (16 \times 10^3 \mathrm{Hz})\)

Simplify the Expression

Simplify the division by dividing the coefficients and subtracting the exponents of 10: Number of channels = \( (1.6 / 16) \times 10^{(12-3)}\) Number of channels = \( 0.1 \times 10^9\)

Convert the Number to Standard Form

Now, we will convert the coefficient 0.1 into standard form by moving the decimal over one place and increasing the exponent by 1: Number of channels = \( 1 \times 10^8\) Thus, the number of communication channels that can be obtained is \(10^8\), which is option (D).

Key concepts

Communication Channels

A communication channel is essentially a pathway that enables data transmission between various devices. In the world of data communication, each channel has a specific bandwidth, which dictates how much information can flow through it in a given timeframe. Think of this like a highway: the wider the highway (or the more lanes it has), the more cars (or data) it can accommodate.
In the context of the problem, we are exploring how many 16 kHz channels fit into a larger bandwidth, which altogether can be used to transmit radio signals. Understanding this allows us to efficiently allocate bandwidth for communication purposes, maximizing the channels we can use without overlap or interference.
By dividing the total available bandwidth by the bandwidth of each communication channel, we determine how many distinct channels can exist within that bandwidth without causing disruption.

Frequency

Frequency refers to how many times a wave oscillates in a given second, measured in Hertz (Hz). For communication signals, frequency is paramount because it defines the signal's position on the electromagnetic spectrum. Different frequencies can coexist without interfering with each other, making this concept crucial when structuring communication systems.
In our exercise, the bandwidth is calculated in terms of frequency—specifically, how many times per second the wave cycles. The total frequency range available, given as 1.6 x 10¹² Hz, allows us to fit a certain number of 16 kHz (16,000 Hz) channels in it. Thus, handling these frequencies efficiently allows for the clear and reliable transmission of information.

Radio Signal

Radio signals use specific frequencies to efficiently transmit data wirelessly over long distances. These signals are part of the radio frequency (RF) portion of the electromagnetic spectrum. They are utilized in various applications such as radio broadcasting, television, and cell phone communication.
The bandwidth of a radio signal determines its capacity to carry information. A broader bandwidth allows a radio signal to transmit more data more quickly. In the exercise, we examine a bandwidth that supports countless communication channels to send and receive data via radio signals simultaneously. This aspect of radio signals is critical in modern technology enabling a multitude of wireless communications.

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers conveniently. It's widely used in sciences and engineering because it makes calculations simpler and more readable, especially when dealing with immense numbers like frequencies.
In the exercise, we express numbers using this notation; for example, 1.6 x 10¹² Hz indicates a frequency of 1,600,000,000,000 Hz. Breaking it down like this using powers of ten makes both computation and comprehension significantly easier. Calculations become much simpler because you can manipulate the exponent and coefficient separately. In this exercise, converting 0.1 to a standard form by adjusting exponents allows us to straightforwardly understand and solve the problem.
Therefore, mastering scientific notation aids in tackling and comprehending vast quantities quickly, particularly useful in fields requiring precise data analysis and computation, such as bandwidth calculations.