Question

A laser pointer used in a lecture hall emits light at \(650 \mathrm{~nm}\). What is the frequency of this radiation? Using Figure 6.4, predict the color associated with this wavelength.

Short answer

The frequency of the laser light is approximately \(4.62 \times 10^{14} \,\text{Hz}\), and the color associated with the given wavelength of \(650 \,\text{nm}\) is red.

Step-by-step solution

Identify the given information

The wavelength, \(λ\), of the laser light is given as \(650\,\text{nm}\) (nanometers).

Convert the wavelength to meters

In order to calculate the frequency in Hertz, it's important to convert the wavelength from nanometers to meters. To do this, remember that \(1 \,\text{nm} = 10^{-9} \,\text{m}\). So, the wavelength of the laser light in meters is: \(λ = 650\,\text{nm} \times \dfrac{1\,\text{m}}{10^9 \,\text{nm}} = 6.5 \times 10^{-7} \,\text{m}\)

Calculate the frequency using the wave speed equation

Now that we have the wavelength in meters, we can calculate the frequency of the laser light using the wave speed equation: \(v = fλ\) The speed of light, \(v\), in vacuum is approximately \(3 \times 10^8 \,\text{m/s}\). So, we can rewrite the equation and solve for frequency, \(f\): \(f = \dfrac{v}{λ}\) \(f = \dfrac{3 \times 10^8 \,\text{m/s}}{6.5 \times 10^{-7} \,\text{m}}\)

Solve for the frequency

Dividing the values, we can find the frequency of the laser light: \(f = \dfrac{3 \times 10^8 \,\text{m/s}}{6.5 \times 10^{-7} \,\text{m}} = 4.62 \times 10^{14}\, \text{Hz}\)

Step 5:Predict the color associated with this wavelength

The frequency of the laser light is found to be approximately \(4.62 \times 10^{14} \,\text{Hz}\). By referring to the provided Figure 6.4, we can predict that the color associated with the given wavelength of \(650 \,\text{nm}\) is red.

Key concepts

Wavelength Conversion

In the realm of physics, understanding how to convert wavelengths from one unit of measure to another is crucial. Light wavelengths are often given in nanometers (nm), which measure the size of light waves on a very fine scale.
Converting wavelengths to meters is essential for various calculations, including determining the frequency of light. To convert from nanometers to meters, you use the conversion:

• 1 nanometer = \(10^{-9}\) meters

So, if you have a wavelength of \(650\,\text{nm}\) for laser light, it would convert to:\[ \lambda = 650\,\text{nm} \times \frac{1\,\text{m}}{10^9\,\text{nm}} = 6.5 \times 10^{-7}\,\text{m} \]This simple step is fundamental when dealing with wave equations in meters, as different units could lead to incorrect results.

Wave Speed Equation

The wave speed equation is a basic but powerful tool used to link the speed, frequency, and wavelength of a wave. The relationship is expressed as:\[ v = f \lambda \]Where:

• \(v\) is the wave speed
• \(f\) is the frequency
• \(\lambda\) is the wavelength

In the context of light, the speed (\(v\)) is usually the speed of light in a vacuum, which is approximately \(3 \times 10^8 \text{m/s}\). To find the frequency (\(f\)), we rearrange the equation to:\[ f = \frac{v}{\lambda} \]Using this formula and our converted wavelength of \(6.5 \times 10^{-7} \text{m}\), we find that the frequency is \(4.62 \times 10^{14} \text{Hz}\). Knowing this relationship is key to understanding how waves behave across different media.

Visible Spectrum Colors

The visible spectrum is the segment of the electromagnetic spectrum that the human eye can perceive. It ranges from approximately 380 nm to 750 nm. Within this range, different wavelengths correspond to different colors. For example:

• Red: 620–750 nm
• Orange: 590–620 nm
• Yellow: 570–590 nm
• Green: 495–570 nm
• Blue: 450–495 nm
• Violet: 380–450 nm

A laser pointer emitting light at \(650 \text{nm}\) falls within the range for red light. Recognizing these color ranges can be handy for predicting the output color of different light sources, especially in practical applications such as lighting and displays.

Laser Light Properties

Laser light is unique due to its coherent properties, meaning the light waves are synchronized in phase and frequency. This coherence provides laser light with distinct characteristics:

• Monochromatic: Lasers emit light of a single wavelength, hence a single color.
• Collimated: Laser light is very directional, maintaining a narrow beam over long distances.
• High Intensity: Because the light is concentrated, laser beams can contain a lot of energy in a concentrated area.

These properties make lasers useful for many applications, from laser pointers and cutting materials to performing delicate surgeries. Understanding the properties of laser light and its behavior at a specific wavelength, like the \(650 \text{nm}\) we discussed, offers insight into its versatile applications in both industrial and scientific fields.