A certain radio wave has a wavelength of 8 × 10-2 m, and a certain gamma ray has a wavelength of 2 × 10-12 m. Which of the following is true?

To determine the relationship between the radio wave and the gamma ray based on their wavelengths, we can use the following understanding of electromagnetic waves:

1. Wavelength and Frequency Relationship: The frequency (f) and wavelength (λ) of electromagnetic waves are inversely related through the speed of light (c), given by the formula: \[ c = f \cdot \lambda \] where \(c\) is approximately \(3 \times 10^8\) m/s.

2. Comparison of Wavelengths:

   • Wavelength of the radio wave: \(λ_{\text{radio}} = 8 \times 10^{-2}\) m (or 0.08 m)
   • Wavelength of the gamma ray: \(λ_{\text{gamma}} = 2 \times 10^{-12}\) m

3. Conclusion:

   • Since the wavelength of the gamma ray (\(2 \times 10^{-12}\) m) is significantly smaller than the wavelength of the radio wave (\(8 \times 10^{-2}\) m), this indicates that the gamma ray has a much higher frequency.
   • Gamma rays have much higher energy than radio waves due to their shorter wavelength.

Therefore, we can conclude that the gamma ray has a higher frequency and higher energy compared to the radio wave.

To find out how many times higher the frequency of the gamma ray is compared to that of the radio wave, we can use the relationship between wavelength and frequency.

1. Calculate the frequency of the radio wave: \[ f_{\text{radio}} = \frac{c}{\lambda_{\text{radio}}} \] Where:

   • \(c \approx 3 \times 10^8\) m/s (speed of light)
   • \(\lambda_{\text{radio}} = 8 \times 10^{-2}\) m

   \[ f_{\text{radio}} = \frac{3 \times 10^8 , \text{m/s}}{8 \times 10^{-2} , \text{m}} = \frac{3 \times 10^8}{0.08} = 3.75 \times 10^9 , \text{Hz} \]

2. Calculate the frequency of the gamma ray: \[ f_{\text{gamma}} = \frac{c}{\lambda_{\text{gamma}}} \] Where:

   • \(\lambda_{\text{gamma}} = 2 \times 10^{-12}\) m

   \[ f_{\text{gamma}} = \frac{3 \times 10^8 , \text{m/s}}{2 \times 10^{-12} , \text{m}} = \frac{3 \times 10^8}{2 \times 10^{-12}} = 1.5 \times 10^{20} , \text{Hz} \]

3. Calculate how many times higher the frequency of the gamma ray is compared to the radio wave: \[ \text{Ratio} = \frac{f_{\text{gamma}}}{f_{\text{radio}}} = \frac{1.5 \times 10^{20} , \text{Hz}}{3.75 \times 10^9 , \text{Hz}} \] \[ \text{Ratio} = \frac{1.5}{3.75} \times 10^{20 - 9} = \frac{1.5}{3.75} \times 10^{11} = 0.4 \times 10^{11} = 4 \times 10^{10} \]

Thus, the frequency of the gamma ray is 40 billion times higher than that of the radio wave.