Problem 6 - University Physics with Modern Physics

Question

There are two categories of ultraviolet light. Ultraviolet A (UVA) has a wavelength ranging from 320 nm to 400 nm. It is necessary for the production of vitamin D. UVB, with a wavelength in vacuum between 280 nm and 320 nm, is more dangerous because it is much more likely to cause skin cancer. (a) Find the frequency ranges of UVA and UVB. (b) What are the ranges of the wave numbers for UVA and UVB?

Step-by-Step Solution

Answer

Frequency ranges: UVA 7.5 - 9.375 x 10^14 Hz, UVB 9.375 - 1.071 x 10^15 Hz. Wave number ranges: UVA 2.5 - 3.125 x 10^6 m^{-1}, UVB 3.125 - 3.571 x 10^6 m^{-1}.

Step 1: Understand the Relationship Between Wavelength and Frequency

The frequency \( f \) of a wave is related to its wavelength \( \lambda \) by the equation \( c = \lambda \cdot f \), where \( c \) is the speed of light in vacuum (approximately \( 3 \times 10^8 \; \text{m/s} \)). We will use this formula to calculate the frequency ranges for both UVA and UVB light.

Step 2: Convert Wavelengths to Meters

Before we calculate frequency, we need to convert the given wavelengths from nanometers to meters since the speed of light is in meters per second. This requires dividing by \( 10^9 \) (since 1 nm = \( 10^{-9} \) m). So, for UVA, the wavelength range is 320 nm to 400 nm, which converts to \( 320 \times 10^{-9} \) m to \( 400 \times 10^{-9} \) m. Similarly, for UVB, the range converts to \( 280 \times 10^{-9} \) m to \( 320 \times 10^{-9} \) m.

Step 3: Calculate Frequency Range for UVA

Using the formula \( f = \frac{c}{\lambda} \) for the converted wavelengths of UVA, calculate the frequency range:- For the minimum wavelength 320 nm: \[ f_\text{min} = \frac{3 \times 10^8}{320 \times 10^{-9}} = 9.375 \times 10^{14} \; \text{Hz} \]- For the maximum wavelength 400 nm: \[ f_\text{max} = \frac{3 \times 10^8}{400 \times 10^{-9}} = 7.5 \times 10^{14} \; \text{Hz} \]Thus, the frequency range for UVA is \( 7.5 \times 10^{14} \) Hz to \( 9.375 \times 10^{14} \) Hz.

Step 4: Calculate Frequency Range for UVB

Use the same formula for UVB:- For the minimum wavelength 280 nm: \[ f_\text{min} = \frac{3 \times 10^8}{280 \times 10^{-9}} = 1.071 \times 10^{15} \; \text{Hz} \]- For the maximum wavelength 320 nm: \[ f_\text{max} = \frac{3 \times 10^8}{320 \times 10^{-9}} = 9.375 \times 10^{14} \; \text{Hz} \]Therefore, the frequency range for UVB is \( 9.375 \times 10^{14} \) Hz to \( 1.071 \times 10^{15} \) Hz.

Step 5: Understand the Concept of Wave Number

The wave number \( k \) is defined as \( k = \frac{1}{\lambda} \), where \( \lambda \) is the wavelength. Since the wavelengths are initially in nanometers, we will convert them to meters for consistency in units (as done previously).

Step 6: Calculate Wave Number Range for UVA

Convert the wavelengths to meters, then find the wave number by taking the reciprocal:- For 400 nm: \( \lambda = 400 \times 10^{-9} \; \text{m} \) \[ \text{Wave number} = \frac{1}{400 \times 10^{-9}} = 2.5 \times 10^6 \; \text{m}^{-1} \]- For 320 nm: \( \lambda = 320 \times 10^{-9} \; \text{m} \) \[ \text{Wave number} = \frac{1}{320 \times 10^{-9}} = 3.125 \times 10^6 \; \text{m}^{-1} \]Thus, the wave number range for UVA is \( 2.5 \times 10^6 \) m\(^{-1}\) to \( 3.125 \times 10^6 \) m\(^{-1}\).

Step 7: Calculate Wave Number Range for UVB

Using the wavelengths for UVB, calculate the wave number:- For 320 nm: \( \lambda = 320 \times 10^{-9} \; \text{m} \) \[ \text{Wave number} = \frac{1}{320 \times 10^{-9}} = 3.125 \times 10^6 \; \text{m}^{-1} \]- For 280 nm: \( \lambda = 280 \times 10^{-9} \; \text{m} \) \[ \text{Wave number} = \frac{1}{280 \times 10^{-9}} = 3.571 \times 10^6 \; \text{m}^{-1} \]Consequently, the wave number range for UVB is \( 3.125 \times 10^6 \) m\(^{-1}\) to \( 3.571 \times 10^6 \) m\(^{-1}\).

Key Concepts

Wavelength

The concept of wavelength is fundamental in understanding waves, including electromagnetic waves such as ultraviolet (UV) light. Wavelength is the distance between two consecutive peaks or troughs in a wave. It is typically measured in meters, but nanometers (nm) are frequently used for light waves.

• UVA light has a wavelength ranging from 320 to 400 nm.
• UVB light ranges from 280 to 320 nm.

Understanding wavelength is crucial because it determines many properties of the wave, including its color in the visible spectrum and its energy. Shorter wavelengths correspond to higher energy and, in the case of UV light, increased potential for biological interactions like sunburn or vitamin D synthesis.

Frequency

Frequency refers to how often the wave cycles occur in a given unit of time. It is measured in hertz (Hz), representing cycles per second. In the realm of light waves, frequency and wavelength share an inverse relationship through the speed of light equation:
\[ c = \lambda \cdot f \]
where:

• \( c \) is the speed of light, approximately \( 3 \times 10^8 \; \text{m/s} \).
• \( \lambda \) is the wavelength in meters.
• \( f \) is the frequency in hertz.

For UVA:

• Minimum frequency at 400 nm: \( 7.5 \times 10^{14} \; \text{Hz} \)
• Maximum frequency at 320 nm: \( 9.375 \times 10^{14} \; \text{Hz} \)

For UVB:

• Minimum frequency at 320 nm: \( 9.375 \times 10^{14} \; \text{Hz} \)
• Maximum frequency at 280 nm: \( 1.071 \times 10^{15} \; \text{Hz} \)

The frequency of a wave influences its energy, with higher frequencies equating to higher energy. This is why UVB waves are more harmful than UVA, even though they both fall under the ultraviolet light classification.

Wave Number

Wave number is a lesser-known but equally important measure in wave analysis. It is the number of wavelengths per unit distance and is calculated as the reciprocal of the wavelength:
\[ k = \frac{1}{\lambda} \]
Wave numbers are usually expressed in \( \text{m}^{-1} \).
For UVA:

• Wave number at 400 nm: \( 2.5 \times 10^6 \; \text{m}^{-1} \)
• Wave number at 320 nm: \( 3.125 \times 10^6 \; \text{m}^{-1} \)

For UVB:

• Wave number at 320 nm: \( 3.125 \times 10^6 \; \text{m}^{-1} \)
• Wave number at 280 nm: \( 3.571 \times 10^6 \; \text{m}^{-1} \)

The wave number is inversely related to wavelength: shorter wavelengths result in higher wave numbers. This measure helps in spectral analysis, providing insight into wave properties and their interactions with materials.

Electromagnetic Spectrum

The electromagnetic spectrum is the full range of all types of electromagnetic radiation. It extends from low-energy radio waves to high-energy gamma rays, with visible light situated in a small segment of the spectrum.

• Ultraviolet light sits just beyond the visible light range and can be divided into UVA and UVB bands.
• UVA and UVB differ mainly in their wavelengths and corresponding impacts.

Understanding the electromagnetic spectrum helps to comprehend why UVB is more harmful: it has shorter wavelengths and higher energies, which can lead to skin damage if not properly protected against. Humans interact with many parts of this spectrum daily, from microwaves used in cooking to visible light required for sight, highlighting the diverse applications and effects of electromagnetic waves.