How much energy does a [tex]$9 x 10^{-8} m$[/tex] wavelength photon have?
A. [tex]$5.96 x 10^{-41} J$[/tex]
B. [tex]$2.21 x 10^{-18} J$[/tex]
C. [tex]$1.988 x 10^{-49} J$[/tex]
D. [tex]$7.36 x 10^{-27} J$[/tex]
Answer (2)
The energy of a photon with a wavelength of 9 × 1 0 − 8 m is calculated to be approximately 2.21 × 1 0 − 18 J. This corresponds to option B in the multiple-choice question. Therefore, the chosen option is B
;
Calculate the frequency of the photon: f = λ c = 9 × 1 0 − 8 m 3 × 1 0 8 m / s .
Calculate the energy of the photon: E = h f = ( 6.626 × 1 0 − 34 J ⋅ s ) × ( 3 1 × 1 0 16 Hz ) .
The energy of the photon is approximately 2.21 × 1 0 − 18 J .
The final answer is: 2.21 × 1 0 − 18 J
Explanation
Problem Analysis and Given Data We are given the wavelength of a photon, λ = 9 × 1 0 − 8 m , and we want to find the energy of the photon. We know that the energy of a photon is related to its frequency by the equation E = h f , where h is Planck's constant ( h = 6.626 × 1 0 − 34 J ⋅ s ) and f is the frequency. We also know that the speed of light, c , is related to the wavelength and frequency by the equation c = λ f , where c = 3 × 1 0 8 m / s .
Calculating the Frequency First, we need to find the frequency of the photon. We can rearrange the equation c = λ f to solve for f : f = λ c Plugging in the given values for c and λ , we get: f = 9 × 1 0 − 8 m 3 × 1 0 8 m / s = 3 1 × 1 0 16 Hz
Calculating the Energy Now that we have the frequency, we can calculate the energy of the photon using the equation E = h f . Plugging in the values for h and f , we get: E = ( 6.626 × 1 0 − 34 J ⋅ s ) × ( 3 1 × 1 0 16 Hz ) = 2.208666... × 1 0 − 18 J
Final Answer and Conclusion Comparing our calculated energy to the given options, we see that the closest answer is 2.21 × 1 0 − 18 J .
Examples
Understanding the energy of photons is crucial in many applications, such as solar energy, medical imaging, and telecommunications. For example, in solar panels, photons from the sun strike the panel and transfer their energy to electrons in the semiconductor material, generating electricity. The efficiency of a solar panel depends on its ability to capture photons of different wavelengths and convert their energy into electricity. Similarly, in medical imaging, X-rays, which are high-energy photons, are used to create images of the inside of the human body. The energy of the X-ray photons determines their ability to penetrate different tissues and create a clear image. In telecommunications, photons are used to transmit information through optical fibers. The energy of the photons determines the amount of information that can be transmitted and the distance over which it can be transmitted.
