Understanding the Wavelength of Light with a Given Frequency: An In-Depth Exploration

Light, a fascinating phenomenon of the electromagnetic spectrum, is characterized by its wavelength and frequency. While these two properties are inversely related, they play a crucial role in determining different aspects of light. This article delves into the mathematics and physics behind calculating the wavelength of light given its frequency, with a focus on a specific example where the frequency is 4.3 x 1014 Hz.

The Formula: Wavelength and Frequency Relationship

When working with light in a vacuum, the wavelength can be determined using the formula:

λ c / f

Where:

λ (lambda) is the wavelength of light c is the speed of light in a vacuum (approximately 3.00 x 108 m/s) f is the frequency of the light wave

This formula is a fundamental relationship in wave theory, linking the speed of light to its wavelength and frequency. It’s widely used in physics and astronomy to analyze various phenomena within the electromagnetic spectrum.

Calculating the Wavelength: A Specific Example

Given a light wave with a frequency of 4.3 x 1014 Hz, we can use the formula to find its wavelength. Let’s walk through the steps:

Identify the values: Speed of light, c 3.00 x 108 m/s Frequency, f 4.3 x 1014 Hz Substitute these values into the formula:

λ (3.00 x 108 m/s) / (4.3 x 1014 Hz)

Perform the division:

λ ≈ (3.00 x 108) / (4.3 x 1014) ≈ 6.98 x 10-7 m

Convert the result from meters to nanometers (1 m 109 nm):

λ ≈ 6.98 x 10-7 m x 109 nm/m ≈ 698 nm

Thus, the wavelength of a light wave with a frequency of 4.3 x 1014 Hz is approximately 698 nm, or 698 nanometers. This corresponds to a deep red light, which falls within the range of visible light.

Wavelength of Waves at Different Frequencies

The relationship between frequency and wavelength is not limited to light. Different types of waves, such as sound waves, have different frequencies and wavelengths depending on the medium they travel through. For instance:

Sound Waves: The speed of sound in air, which is about 343 m/s, would give a different wavelength for the same frequency of 4.3 x 1014 Hz. Calculating this:

λ (343 m/s) / (4.3 x 1014 Hz) ≈ 8.0 x 10-12 m or 800 pm (picometers)

These sound waves would have a very short wavelength since they travel much slower in air than light. Sound in Other Media: Sound travels faster in denser mediums like water or steel. Therefore, the wavelength would be shorter in these media. For example, in water:

λ (1480 m/s) / (4.3 x 1014 Hz) ≈ 3.44 x 10-5 m or 34.4 μm

And in steel:

λ (5960 m/s) / (4.3 x 1014 Hz) ≈ 1.38 x 10-9 m or 1.38 nm

These examples illustrate just how the physical properties of the medium can affect the behavior of waves, such as their frequency and wavelength.

Conclusion

Understanding the relationship between wavelength and frequency is essential in various fields, from astronomy to telecommunications. The formula λ c / f serves as a powerful tool for calculating the wavelength of light or other waves, given their frequency. This simple yet profound relationship underpins much of our understanding of the electromagnetic spectrum and wave propagation in different mediums.

Additional Resources

For a deeper dive into the topic, consider exploring the Speed of Light on Wikipedia or Physics Classroom's tutorial on Wavelength and Frequency.