Problem 8 - Physics

Question

Obtain the wavelengths in vacuum for (a) blue light whose frequency is \(6.34 \times 10^{14} \mathrm{~Hz}\) and (b) orange light whose frequency is \(4.95 \times 10^{14} \mathrm{~Hz}\). Express your answers in nanometers \(\left(1 \mathrm{nm}=10^{-9} \mathrm{~m}\right)\)

Step-by-Step Solution

Answer

Blue light: 473 nm, Orange light: 606 nm.

Step 1: Understanding the Relationship Between Wavelength, Frequency, and Speed of Light

The speed of light in a vacuum is denoted by the symbol \(c\) and is approximately equal to \(3.00 \times 10^8 \) meters per second. The relationship between the speed of light \(c\), its frequency \(f\), and its wavelength \(\lambda\) is given by the formula: \(c = f \times \lambda\). We will use this formula to find the wavelength by rearranging it to \(\lambda = \frac{c}{f}\).

Step 2: Calculating the Wavelength for Blue Light

The frequency of blue light is given as \(6.34 \times 10^{14} \mathrm{~Hz}\). Using the formula \(\lambda = \frac{c}{f}\), where \(c = 3.00 \times 10^8 \mathrm{~m/s}\), the wavelength is:\[\lambda_{\text{blue}} = \frac{3.00 \times 10^8}{6.34 \times 10^{14}} \approx 4.73 \times 10^{-7} \text{ meters}\]Since the final answer is needed in nanometers, we convert meters into nanometers:\[\lambda_{\text{blue}} = 4.73 \times 10^{-7} \text{ meters} \times 10^{9} \text{ nm/m} = 473 \text{ nm}\].

Step 3: Calculating the Wavelength for Orange Light

The frequency of orange light is given as \(4.95 \times 10^{14} \mathrm{~Hz}\). Using the formula \(\lambda = \frac{c}{f}\), the wavelength is:\[\lambda_{\text{orange}} = \frac{3.00 \times 10^8}{4.95 \times 10^{14}} \approx 6.06 \times 10^{-7} \text{ meters}\]Converting this to nanometers gives:\[\lambda_{\text{orange}} = 6.06 \times 10^{-7} \text{ meters} \times 10^{9} \text{ nm/m} = 606 \text{ nm}\].

Key Concepts

Frequency

Frequency refers to the number of times a wave oscillates or cycles through a point in one second. It is typically measured in hertz (Hz), where 1 Hz equals one cycle per second. In the context of light, frequency determines the color of light within the visible spectrum.
For instance, in our previous exercise, the frequency for blue light was given as \(6.34 \times 10^{14} \text{ Hz}\) and for orange light as \(4.95 \times 10^{14} \text{ Hz}\). This means blue light waves oscillate more times per second than orange light waves, making blue light a higher frequency light compared to orange light.
Understanding frequency is crucial because it is directly related to a wave's energy and its wavelength. A high-frequency wave has a shorter wavelength, which means its peaks are closer together. This inverse relationship is essential for calculating wavelengths in exercises involving light and the electromagnetic spectrum.

Speed of Light

The speed of light in a vacuum is a fundamental constant in physics, represented by the symbol \(c\). It is approximately \(3.00 \times 10^8 \text{ meters per second}\). This constant is crucial because it establishes the maximum speed at which all energy, matter, and information in the universe can travel.

• It is the speed at which electromagnetic waves, including visible light, travel in a vacuum.
• The speed of light remains consistent in a vacuum but can vary when light enters different mediums such as air, water, or glass.

In the context of calculating wavelengths, the speed of light helps bridge the relationship between frequency and wavelength through the formula \(c = f \times \lambda \).
When rearranged to \(\lambda = \frac{c}{f}\), it allows us to calculate the wavelength knowing the frequency, which is critical in the exercise on light spectra.

Light Spectrum

The light spectrum refers to the range of all types of electromagnetic radiation. We typically think about visible light, which is the portion of the spectrum that can be detected by the human eye.
The visible light spectrum includes a set of colors, each associated with a particular range of wavelengths and frequencies: red, orange, yellow, green, blue, indigo, and violet. For instance, blue and orange are both part of this spectrum but appear as different colors due to their differing frequencies and wavelengths.

• Blue light has a shorter wavelength (around 450-495 nm) and a higher frequency, giving it more energy compared to red or orange light.
• Orange light has a longer wavelength (around 590-620 nm) and a lower frequency than blue light.

Light behaves both as a particle and a wave, and understanding its spectrum facilitates the exploration of various phenomena in physics, such as diffraction, refraction, and interference.