Question

How much bandwidth is there in \(0.1\) microns of spectrum at a wavelength of 1 micron?

Short answer

The bandwidth is approximately \(6.06 \times 10^{13}\) Hz.

Step-by-step solution

Understanding the Problem

The problem asks us to find the bandwidth, expressed in terms of frequency, when the wavelength changes by 0.1 microns centered around a 1 micron wavelength. This is essentially the partial change in frequency corresponding to the given range of wavelength.

Using the Basic Formula

The relationship between speed, frequency, and wavelength is given by the formula: \[ c = \lambda \cdot f \] where \( c \) is the speed of light (approximately \(3 \times 10^8\) meters per second), \( \lambda \) is the wavelength in meters, and \( f \) is the frequency in hertz. Here, we will use this formula to express frequency \( f \) in terms of wavelength \( \lambda \).

Calculating Frequency at 1 Micron

Convert 1 micron to meters: \[ 1 \text{ micron} = 1 \times 10^{-6} \text{ meters} \] Using the formula \( c = \lambda \cdot f \), solve for \( f \): \[ f = \frac{c}{\lambda} \] Substitute \( \lambda = 1 \times 10^{-6} \) meters: \[ f = \frac{3 \times 10^8}{1 \times 10^{-6}} = 3 \times 10^{14} \text{ Hz} \] This is the frequency at a wavelength of 1 micron.

Calculating Frequency at 1.1 Microns

Use the same method with \( \lambda = 1.1 \times 10^{-6} \) meters: \[ f = \frac{3 \times 10^8}{1.1 \times 10^{-6}} \approx 2.727 \times 10^{14} \text{ Hz} \] This is the frequency at a wavelength of 1.1 microns.

Calculating Frequency at 0.9 Microns

Similarly, for \( \lambda = 0.9 \times 10^{-6} \) meters: \[ f = \frac{3 \times 10^8}{0.9 \times 10^{-6}} \approx 3.333 \times 10^{14} \text{ Hz} \] This is the frequency at a wavelength of 0.9 microns.

Finding the Bandwidth

The bandwidth \( \Delta f \) is the difference between the frequencies at 0.9 microns and 1.1 microns: \[ \Delta f = 3.333 \times 10^{14} - 2.727 \times 10^{14} = 0.606 \times 10^{14} \text{ Hz} \] Which is: \[ 6.06 \times 10^{13} \text{ Hz} \] This is the bandwidth for the 0.1 micron change in the spectrum around the 1 micron wavelength.

Key concepts

Wavelength

Wavelength is a measure of the distance between two consecutive peaks (or troughs) in a wave, essentially the length of one wave cycle. It is usually denoted by the Greek letter lambda (\( \lambda \)) and is measured in meters for most scientific applications. However, when dealing with light and other forms of electromagnetic radiation, wavelengths can become extremely small. This is why microns, which are millionths of a meter (\( 1 \text{ micron} = 1 \times 10^{-6} \text{ meters} \)), are often used in calculations.

The concept of wavelength is critical in understanding how waves propagate through different mediums. When you know a wave's wavelength, you can find other important properties of the wave, such as its frequency and speed. Particularly in electromagnetic waves, like light, the wavelength determines the wave's energy and type. Shorter wavelengths generally carry more energy and may fall into categories like ultraviolet or x-rays, whereas longer wavelengths include visible and infrared light.

Frequency

Frequency is the number of complete wave cycles that pass a point in one second. It is denoted by \( f \) and is measured in hertz (Hz), where one hertz equals one cycle per second.

In the electromagnetic spectrum, frequency plays a crucial role in defining the type of radiation. Higher frequencies correspond to more energetic waves like gamma rays or X-rays, while lower frequencies are associated with less energetic waves like radio waves. Relationship between frequency and wavelength is an inverse one, meaning when the wavelength of a wave increases, its frequency decreases, and vice versa. Thus, these properties are interdependent and are used together to describe waves completely.

• Frequency tells us how fast the wave oscillates.
• It is crucial for defining the energy and type of waves within the spectrum.

Understanding frequency helps us when solving physics and engineering problems. For instance, determining the bandwidth of a wave, which is a range of frequencies, is particularly useful in communication technologies.

Speed of light

The speed of light, often denoted by \( c \), is a fundamental constant in physics. It is the speed at which all electromagnetic waves travel in a vacuum and is approximately \( 3 \times 10^8 \text{ meters per second} \). This speed is invariant and plays a pivotal role in various physics equations and theories, like Einstein's theory of relativity.

The equation \( c = \lambda \cdot f \) expresses the relationship between the speed of light, wavelength, and frequency. This formula highlights how these properties are interlinked in electromagnetic waves:

• If you know two of these properties, you can easily calculate the third.
• This relationship helps in assessing other wave characteristics, like energy.
• It is foundational for calculations involving optical and radio technologies.

Understanding light's speed is crucial not only in physics, but also in numerous applications such as telecommunications, where it is vital to calculate data transmission rates efficiently and accurately.