Problem 4 - University Physics with Modern Physics

Question

There are two categories of ultraviolet light. Ultraviolet A (UVA) has a wavelength ranging from 320 \(\mathrm{nm}\) to 400 \(\mathrm{nm}\) . It is not so harmful to the skin and is necessary for the production of vitamin D. UVB, with a wavelength between 280 \(\mathrm{nm}\) and 320 \(\mathrm{mm}\) , is much more dangerous because it causes 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

(a) UVA frequencies: 7.5 to 9.375 x 10^14 Hz; UVB frequencies: 9.375 x 10^14 to 1.071 x 10^15 Hz. (b) UVA wave numbers: 2.5 to 3.125 x 10^6 m^-1; UVB wave numbers: 3.125 to 3.571 x 10^6 m^-1.

Step 1: Understanding the Relationship Between Wavelength and Frequency

To find the frequency, use the equation \( c = \lambda u \), where \( c \) is the speed of light (approximately \( 3 \times 10^8 \, \text{m/s} \)), \( \lambda \) is the wavelength in meters, and \( u \) is the frequency in Hz. Re-arranging gives \( u = \frac{c}{\lambda} \).

Step 2: Convert Wavelengths from Nanometers to Meters

For Ultraviolet A (UVA), convert the range: 320 nm to 400 nm into meters by remembering that \( 1 \, \text{nm} = 10^{-9} \, \text{m} \). Thus: \( 320 \, \text{nm} = 320 \times 10^{-9} \, \text{m} \) and \( 400 \, \text{nm} = 400 \times 10^{-9} \, \text{m} \). Similarly, for UVB: \( 280 \, \text{nm} = 280 \times 10^{-9} \, \text{m} \) and \( 320 \, \text{nm} = 320 \times 10^{-9} \, \text{m} \).

Step 3: Calculate Frequency Ranges for UVA

UVA's frequency range is calculated using the wavelengths converted to meters. For 320 nm: \( u = \frac{3 \times 10^8}{320 \times 10^{-9}} = 9.375 \times 10^{14} \, \text{Hz} \). For 400 nm: \( u = \frac{3 \times 10^8}{400 \times 10^{-9}} = 7.5 \times 10^{14} \, \text{Hz} \). Therefore, the frequency range for UVA is from \( 7.5 \times 10^{14} \, \text{Hz} \) to \( 9.375 \times 10^{14} \, \text{Hz} \).

Step 4: Calculate Frequency Ranges for UVB

Using the wavelengths for UVB: For 280 nm: \( u = \frac{3 \times 10^8}{280 \times 10^{-9}} = 1.071 \times 10^{15} \, \text{Hz} \). For 320 nm: \( u = \frac{3 \times 10^8}{320 \times 10^{-9}} = 9.375 \times 10^{14} \, \text{Hz} \). Therefore, the frequency range for UVB is from \( 9.375 \times 10^{14} \, \text{Hz} \) to \( 1.071 \times 10^{15} \, \text{Hz} \).

Step 5: Understanding Wave Number

The wave number, \( \bar{k} \), is defined as the reciprocal of the wavelength \( \bar{k} = \frac{1}{\lambda} \), typically expressed in \( \text{m}^{-1} \).

Step 6: Calculate Wave Number Ranges for UVA

Using the UV wavelengths in meters, calculate for UVA: At 320 nm, \( \bar{k} = \frac{1}{320 \times 10^{-9}} = 3.125 \times 10^{6} \, \text{m}^{-1} \) and at 400 nm, \( \bar{k} = \frac{1}{400 \times 10^{-9}} = 2.5 \times 10^{6} \, \text{m}^{-1} \). Hence, UVA's wave number range is from \( 2.5 \times 10^{6} \, \text{m}^{-1} \) to \( 3.125 \times 10^{6} \, \text{m}^{-1} \).

Step 7: Calculate Wave Number Ranges for UVB

Similarly, for UVB: At 280 nm, \( \bar{k} = \frac{1}{280 \times 10^{-9}} = 3.571 \times 10^{6} \, \text{m}^{-1} \) and at 320 nm, \( \bar{k} = \frac{1}{320 \times 10^{-9}} = 3.125 \times 10^{6} \, \text{m}^{-1} \). Thus, the wave number range for UVB is from \( 3.125 \times 10^{6} \, \text{m}^{-1} \) to \( 3.571 \times 10^{6} \, \text{m}^{-1} \).

Key Concepts

Frequency Calculation

Frequency calculation is an essential concept when understanding waves, especially in physics. The frequency of a wave describes how often the wave oscillates in a given time period. For electromagnetic waves like ultraviolet light, we use the speed of light and the wavelength to calculate frequency.

• The formula to find the frequency (\( u \)) is given by \( u = \frac{c}{\lambda} \), where \( c = 3 \times 10^8 \, \text{m/s} \) is the speed of light and \( \lambda \) is the wavelength.
• Shorter wavelengths correspond to higher frequencies.

To determine the frequency range for UV light like UVA or UVB, first convert the wavelength from nanometers to meters, since the speed of light is in meters per second. You can achieve this by using the conversion: \( 1 \, \text{nm} = 10^{-9} \, \text{m} \). Understanding frequency helps us determine how much energy is carried by a wave, since higher frequencies mean higher energy which can assist in explaining the differing effects of UVA and UVB.

Wavelength

Wavelength is a fundamental aspect of wave science. It describes the distance between two consecutive peaks of a wave. In the case of ultraviolet (UV) light, wavelength plays a crucial role in determining the light's potential effects and uses.

• UV light is categorized into two main types based on wavelength: Ultraviolet A (UVA) with wavelengths from 320 to 400 \( \text{nm} \) and Ultraviolet B (UVB) ranging from 280 to 320 \( \text{nm} \).
• A longer wavelength indicates less energy per photon, whereas a shorter wavelength denotes more energy.

These wavelength ranges are significant. While UVA aids in Vitamin D production and is generally less harmful, UVB is more energetic and poses a greater risk for skin damage and cancer. By understanding the wavelength, one can easily infer the energy of UV radiation and its implications.

Wave Number

Wave number is a useful way to represent waves, especially within the field of spectroscopy. It provides a measure of the number of wave cycles in a given length unit, typically inverse meters (\( \text{m}^{-1} \)).

• The wave number (\( \bar{k} \)) is calculated as \( \bar{k} = \frac{1}{\lambda} \).
• It is directly related to wavelength; a shorter wavelength results in a higher wave number.

For UV radiation, calculating the wave number gives insight into the energy characteristics of UVA and UVB. For instance, a higher wave number for UVB compared to UVA additionally confirms its greater potential to cause skin damage. Wave numbers are often used in spectroscopy since they tend to align well with the scales and units commonly used in those applications.