Question

Which wavelength has the lower frequency: \(450 \mathrm{nm}\) or \(550 \mathrm{nm} ?\)

Short answer

The wavelength 550 nm has the lower frequency.

Step-by-step solution

Understand the Relationship

The frequency () and wavelength (\(\lambda)\) of a wave are inversely related using the speed of light equation: \[ c = \lambda \cdot u \]where \(c\) is the speed of light, approximately \(3 \times 10^8 \text{ m/s}\). Thus, for a fixed speed of light, wavelength and frequency are inversely related.

Convert Wavelengths to Meters

We know that 1 nm = \(1 \times 10^{-9}\) m. So, convert 450 nm and 550 nm to meters:\[ 450 \text{ nm} = 450 \times 10^{-9} \text{ m} \]\[ 550 \text{ nm} = 550 \times 10^{-9} \text{ m} \]

Calculate Frequency of Each Wavelength

Using the equation \(u = \frac{c}{\lambda}\), calculate the frequency for each wavelength.For 450 nm:\[ u_{450} = \frac{3 \times 10^8}{450 \times 10^{-9}} \]And for 550 nm:\[ u_{550} = \frac{3 \times 10^8}{550 \times 10^{-9}} \]

Compare the Frequencies

Calculate the actual values. The smaller wavelength (450 nm) results in a higher frequency, while the larger wavelength (550 nm) results in a lower frequency.\[ u_{450} \approx 6.67 \times 10^{14} \text{ Hz} \]\[ u_{550} \approx 5.45 \times 10^{14} \text{ Hz} \]

Conclusion

Comparing the calculated frequencies, the wavelength of 550 nm has the lower frequency compared to 450 nm.

Key concepts

Understanding the Speed of Light

The speed of light is a fundamental constant that is crucial in the study of wave behavior, particularly in the context of electromagnetic waves. It is represented by the symbol \( c \) and is approximately equal to \( 3 \times 10^8 \text{ m/s} \). This constant speed is significant because it dictates how quickly electromagnetic waves, such as light, travel in a vacuum.
Light speed links wavelength and frequency in wave calculations. No matter the wavelength, light's speed in a vacuum remains the same, which is why it is crucial for calculating wave properties. Understanding the speed of light allows us to determine the relationship between two key properties of a wave: its wavelength and frequency, using the formula \( c = \lambda \cdot u \).
Key Points:

• Speed of light (\( c \)) is always \( 3 \times 10^8 \text{ m/s} \) in a vacuum.
• This speed is used to relate wavelength (\( \lambda \)) and frequency (\( u \)) of a light wave.
• Independent of the waveform, \( c = \lambda \cdot u \) holds true for wave calculations.

Having a constant speed simplifies calculations and helps predict how waves behave.

Inverse Proportionality Explained

The concept of inverse proportionality is central when examining the relationship between wavelength and frequency. Inverse proportionality means that, as one variable increases, the other decreases. This principle is evident in the equation \( c = \lambda \cdot u \), where an increase in wavelength (\( \lambda \)) results in a decrease in frequency (\( u \)), provided the speed of light \( c \) remains constant.
Understanding this relationship helps to clarify why different wavelengths of light do not all have the same frequency. When you calculate frequency using \( u = \frac{c}{\lambda} \), the inverse relationship becomes evident.
Key Observations Include:

• If \( \lambda \) (wavelength) increases, \( u \) (frequency) decreases.
• This relationship can be seen in the simple example of 450 nm versus 550 nm wavelengths.
• Longer wavelengths have lower frequencies; shorter wavelengths have higher frequencies.

By recognizing inverse proportionality, predictions about wave behavior become intuitive, allowing for more straightforward conclusions in scientific studies.

Performing Wave Calculations

Wave calculations rely heavily on the interplay between wavelength, frequency, and the speed of light. Calculating frequency from the wavelength requires the formula \( u = \frac{c}{\lambda} \). Let's break it down with examples:
Convert nanometers to meters for ease of calculation, since the speed of light is in meters per second. For example, 450 nm converts to \( 450 \times 10^{-9} \text{ m} \) and 550 nm converts to \( 550 \times 10^{-9} \text{ m} \). Plugging these into the formula gives:
For a 450 nm wavelength: \[ u_{450} = \frac{3 \times 10^8}{450 \times 10^{-9}} \] For a 550 nm wavelength: \[ u_{550} = \frac{3 \times 10^8}{550 \times 10^{-9}} \]
Observations from Calculations:

• A smaller wavelength (450 nm) results in a higher frequency (~\( 6.67 \times 10^{14} \text{ Hz} \)).
• A larger wavelength (550 nm) results in a lower frequency (~\( 5.45 \times 10^{14} \text{ Hz} \)).
• Understanding conversion and calculation is essential for accurate interpretation of wave behavior.

Learning to perform these calculations solidifies understanding and assists in various scientific applications, from telecommunications to understanding the color of light.