Question

calculate the frequency of each wavelength of electromagnetic
radiation:
a. 632.8 nm (wavelength of red light from helium - neon laser)
b. 503 nm (wavelength of maximum solar radiation)
c. 0.052 nm (a wavelength contained in medical x - rays)

Explanation:

To calculate the frequency of electromagnetic radiation, we use the formula \( c = \lambda
u \), where \( c \) is the speed of light (\( c = 3.00 \times 10^8 \, \text{m/s} \)), \( \lambda \) is the wavelength, and \(
u \) is the frequency. We can rearrange this formula to solve for frequency: \(
u = \frac{c}{\lambda} \). Note that we need to convert the wavelength from nanometers (nm) to meters (m) since \( 1 \, \text{nm} = 10^{-9} \, \text{m} \).

Part (a): Wavelength \( \lambda = 632.8 \, \text{nm} \)

Step 1: Convert wavelength to meters

\( \lambda = 632.8 \, \text{nm} \times \frac{10^{-9} \, \text{m}}{1 \, \text{nm}} = 632.8 \times 10^{-9} \, \text{m} = 6.328 \times 10^{-7} \, \text{m} \)

Step 2: Calculate frequency using \(

u = \frac{c}{\lambda} \)
\(
u = \frac{3.00 \times 10^8 \, \text{m/s}}{6.328 \times 10^{-7} \, \text{m}} \approx 4.74 \times 10^{14} \, \text{Hz} \)

Part (b): Wavelength \( \lambda = 503 \, \text{nm} \)

Step 1: Convert wavelength to meters

\( \lambda = 503 \, \text{nm} \times \frac{10^{-9} \, \text{m}}{1 \, \text{nm}} = 503 \times 10^{-9} \, \text{m} = 5.03 \times 10^{-7} \, \text{m} \)

Step 2: Calculate frequency using \(

u = \frac{c}{\lambda} \)
\(
u = \frac{3.00 \times 10^8 \, \text{m/s}}{5.03 \times 10^{-7} \, \text{m}} \approx 5.96 \times 10^{14} \, \text{Hz} \)

Part (c): Wavelength \( \lambda = 0.052 \, \text{nm} \)

Step 1: Convert wavelength to meters

\( \lambda = 0.052 \, \text{nm} \times \frac{10^{-9} \, \text{m}}{1 \, \text{nm}} = 0.052 \times 10^{-9} \, \text{m} = 5.2 \times 10^{-11} \, \text{m} \)

Step 2: Calculate frequency using \(

u = \frac{c}{\lambda} \)
\(
u = \frac{3.00 \times 10^8 \, \text{m/s}}{5.2 \times 10^{-11} \, \text{m}} \approx 5.77 \times 10^{18} \, \text{Hz} \)

Final Answers:

• (a) \( \approx 4.74 \times 10^{14} \, \text{Hz} \)
• (b) \( \approx 5.96 \times 10^{14} \, \text{Hz} \)
• (c) \( \approx 5.77 \times 10^{18} \, \text{Hz} \)

Answer:

To calculate the frequency of electromagnetic radiation, we use the formula \( c = \lambda
u \), where \( c \) is the speed of light (\( c = 3.00 \times 10^8 \, \text{m/s} \)), \( \lambda \) is the wavelength, and \(
u \) is the frequency. We can rearrange this formula to solve for frequency: \(
u = \frac{c}{\lambda} \). Note that we need to convert the wavelength from nanometers (nm) to meters (m) since \( 1 \, \text{nm} = 10^{-9} \, \text{m} \).

Part (a): Wavelength \( \lambda = 632.8 \, \text{nm} \)

Step 1: Convert wavelength to meters

\( \lambda = 632.8 \, \text{nm} \times \frac{10^{-9} \, \text{m}}{1 \, \text{nm}} = 632.8 \times 10^{-9} \, \text{m} = 6.328 \times 10^{-7} \, \text{m} \)

Step 2: Calculate frequency using \(

u = \frac{c}{\lambda} \)
\(
u = \frac{3.00 \times 10^8 \, \text{m/s}}{6.328 \times 10^{-7} \, \text{m}} \approx 4.74 \times 10^{14} \, \text{Hz} \)

Part (b): Wavelength \( \lambda = 503 \, \text{nm} \)

Step 1: Convert wavelength to meters

\( \lambda = 503 \, \text{nm} \times \frac{10^{-9} \, \text{m}}{1 \, \text{nm}} = 503 \times 10^{-9} \, \text{m} = 5.03 \times 10^{-7} \, \text{m} \)

Step 2: Calculate frequency using \(

u = \frac{c}{\lambda} \)
\(
u = \frac{3.00 \times 10^8 \, \text{m/s}}{5.03 \times 10^{-7} \, \text{m}} \approx 5.96 \times 10^{14} \, \text{Hz} \)

Part (c): Wavelength \( \lambda = 0.052 \, \text{nm} \)

Step 1: Convert wavelength to meters

\( \lambda = 0.052 \, \text{nm} \times \frac{10^{-9} \, \text{m}}{1 \, \text{nm}} = 0.052 \times 10^{-9} \, \text{m} = 5.2 \times 10^{-11} \, \text{m} \)

Step 2: Calculate frequency using \(

u = \frac{c}{\lambda} \)
\(
u = \frac{3.00 \times 10^8 \, \text{m/s}}{5.2 \times 10^{-11} \, \text{m}} \approx 5.77 \times 10^{18} \, \text{Hz} \)

Final Answers:

• (a) \( \approx 4.74 \times 10^{14} \, \text{Hz} \)
• (b) \( \approx 5.96 \times 10^{14} \, \text{Hz} \)
• (c) \( \approx 5.77 \times 10^{18} \, \text{Hz} \)