Question

A 42.58 -MHz photon is needed to produce nuclear magnetic resonance in free protons in a magnetic field of \(1.000 \mathrm{~T}\). What is the wavelength of the photon, its energy, and the region of the spectrum in which it lies? Could it be harmful to the human body?

Short answer

Answer: The wavelength of the 42.58-MHz photon is 7.05 x 10^-3 m, its energy is 2.82 x 10^-26 J, it is part of the radio waves region of the spectrum, and it is not harmful to the human body.

Step-by-step solution

Calculate the wavelength of the photon

Using the relationship between speed of light, wavelength, and frequency, we can find the wavelength of the photon: $$\lambda = \frac{c}{\nu}$$ The speed of light, \(c = 3.00 \times 10^8 \frac{m}{s}\). The frequency, \(\nu = 42.58\times 10^6 Hz\). Substitute the values to find the wavelength: $$\lambda = \frac{3.00\times 10^8\frac{m}{s}}{42.58\times 10^6 Hz}$$ $$\lambda = 7.05 \times 10^{-3} m$$

Calculate the energy of the photon

Next, we will find the energy of the photon using the relationship between energy, Planck's constant, and frequency: $$E = h\nu$$ Planck's constant, \(h = 6.626\times 10^{-34} J\cdot s\). The frequency, \(\nu = 42.58\times 10^6 Hz\). Substitute the values to find the energy: $$E = (6.626\times 10^{-34}J\cdot s)(42.58\times 10^6 Hz)$$ $$E = 2.82 \times 10^{-26} J$$

Identify the region of the spectrum

The wavelength obtained is \(7.05\times 10^{-3} m\). This wavelength lies in the radio waves region of the electromagnetic spectrum.

Analyze if the photon is harmful to the human body

The photon is in the radio waves region of the electromagnetic spectrum, which is a non-ionizing radiation. Non-ionizing radiations generally do not have enough energy to remove tightly bound electrons from atoms and therefore are not harmful to the human body. As a result, the 42.58-MHz photon would not be harmful to the human body. In conclusion, for the 42.58-MHz photon, the wavelength is \(7.05\times 10^{-3} m\), its energy is \(2.82\times 10^{-26} J\), it is part of the radio waves region of the spectrum, and it is not harmful to the human body.

Key concepts

Wavelength Calculation

Understanding the calculation of wavelength is crucial when it comes to analyzing phenomena such as nuclear magnetic resonance (NMR). The wavelength of a photon can be determined using the relation \( \lambda = \frac{c}{u} \), where \( \lambda \) is the wavelength, \( c \) is the speed of light in a vacuum (approximately \(3.00 \times 10^8 \frac{m}{s}\)), and \( u \) is the frequency of the electromagnetic wave.

Given the frequency as 42.58 MHz, or \(42.58 \times 10^6 Hz\), we can substitute these values into our equation to find the wavelength. Such calculations reveal the wave aspect of electromagnetic radiation, showing that its properties are linked with both particles (photons) and waves.

In the context of NMR, the precise wavelength corresponding to a specific frequency is essential to ensure that the photon energy matches the energy gap between nuclear spin alignments in a magnetic field. It fosters an understanding of how energy is transferred at an atomic level, influencing the movement and orientation of nuclear particles.

Photon Energy

Photon energy is quantified by the equation \( E = hu \) where \( E \) represents the energy of the photon, \( h \) is the Planck's constant \(6.626 \times 10^{-34} J\cdot s\), and \( u \) is the frequency of the photon's associated electromagnetic wave.

This equation intertwines the realms of quantum mechanics and electromagnetism, indicating that photons carry energy proportional to their frequency. When we apply the frequency of 42.58 MHz, the photon energy calculation results in \(2.82 \times 10^{-26} J\), suggesting that photons with lower frequencies, like ones used in NMR, carry less energy compared to high-frequency photons such as gamma rays or X-rays.

This concept is foundational to safety in medical imaging and other applications, as it helps us infer the potential for biological damage. Since the energy of an NMR photon isn't sufficient to ionize atoms, it is deemed non-harmful to human tissue, which is fundamental knowledge for both the understanding of NMR safety and its practical implications.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all forms of electromagnetic radiation, ranging from the very long wavelengths of radio waves to the incredibly short wavelengths of gamma rays. Each type of radiation within the spectrum, such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays, has its characteristic wavelength and frequency range.

Radio waves, which include the wavelength \(7.05 \times 10^{-3} m\) calculated for the 42.58 MHz photon used in NMR, are at the long-wavelength end of the spectrum. They are well-known for their application in broadcasting, communication, and medical imaging. Distinguishing between these regions of the spectrum is critical for various technologies and for understanding the interaction of electromagnetic radiation with matter.

Furthermore, identifying the spectrum region informs us about possible risks or applications – long wavelengths such as those in radio waves are typically harmless to biological organisms, while shorter wavelengths, like ultraviolet and beyond, can cause cellular damage. This knowledge is integral to the safe and effective use of electromagnetic radiation in both scientific and everyday contexts.