Question

To transmit \(1 \mathrm{KHz}\) frequency signal, wave length of the signal should be \(\quad \mathrm{m}\). (A) \(3 \times 10^{3}\) (B) \(3 \times 10^{5}\) (C) \(3 \times 10^{7}\) (D) \(3 \times 10^{10}\)

Short answer

The wavelength of a 1 kHz frequency signal is \(3 \times 10^{5}\) meters (Option B).

Step-by-step solution

Convert frequency to Hz

Given Frequency (f) = 1 kHz We should convert this to Hz by multiplying by 1000: f = 1 * 1000 = 1000 Hz

Calculate the wavelength

Now we'll use the formula to calculate the wavelength: Wavelength (λ) = Speed of light (c) / Frequency (f) The speed of light (c) is approximately 3 x 10^8 meters per second (m/s): λ = (3 x 10^8) / 1000

Simplify the expression

Simplify the expression: λ = (3 x 10^8) / 10^3 = 3 x 10^5 m

Select the correct option

The calculated value of the wavelength is \(3 \times 10^{5}\) m. This means the correct option is (B).

Key concepts

Frequency

Frequency is a fundamental concept in the realm of physics, specifically in wave mechanics. It describes how often the peaks of a wave pass a particular point over a given period. Frequency is generally measured in units called Hertz (Hz), which equal one cycle per second.

• For example, a signal transmitting at 1 KHz is completing 1,000 cycles per second.
• In many applications, such as radio broadcasting, frequency indicates the specific station or signal type.

Understanding frequency is critical for determining the properties of waves, which influence how we perceive them and how they interact with materials. Converting to the standard unit of Hertz often simplifies calculations related to wave properties like wavelength.

Wavelength Calculation

Wavelength, denoted by the Greek letter lambda (\( \lambda \)), is the distance between consecutive crests of a wave. It is an essential characteristic of wave phenomena and is crucial in describing a wave's behavior and energy.

• When calculating wavelength, the formula \( \lambda = \frac{c}{f} \) is typically used, where \( c \) is the speed of light and \( f \) is the frequency.
• In our example, converting 1 KHz to 1000 Hz allows us to use this formula to find that \( \lambda = 3 \times 10^5 \) meters.

Wavelength is inversely proportional to frequency, meaning that as one increases, the other decreases. This implies that longer wavelengths correspond to lower frequencies, and vice versa.

Speed of Light

The speed of light, denoted as \( c \), is one of the universal constants of nature. It determines how fast light travels through a vacuum and is approximately \( 3 \times 10^8 \) meters per second.

• This speed is vital for both scientific calculations and technological applications, such as telecommunications.
• In the formula \( \lambda = \frac{c}{f} \), the speed of light is the constant that bridges the relationship between frequency and wavelength.

Understanding the speed of light helps us grasp how electromagnetic waves propagate, and it plays a crucial role in theories like relativity. In everyday technologies, this constant allows for efficient design and analysis of communication systems that rely on signal transmission.

Unit Conversion

Unit conversion is an essential mathematical skill in science, helping ensure that measurements are utilized appropriately in calculations. In waveform analysis, converting between units like KHz to Hz facilitates a clear understanding.

• Converting 1 KHz to Hz equals multiplying by 1,000, leading to 1,000 Hz.
• This same principle applies to many other units, such as kilometers to meters or hours to seconds.

Ensuring measurements are in the correct units before performing calculations avoids errors and provides clarity. It standardizes data so that it aligns with universal measurements, making it easier to share, compare, and apply results accurately across different fields of study.