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Chapter 40: Problem 54

Consider a 42.58 -MHz photon needed to produce NMR transition 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

Expert verified
Answer: The wavelength of the photon is 7.045 × 10^-3 m (7.045 mm), its energy is 2.823 × 10^-26 J, it belongs to the radio wave region of the electromagnetic spectrum, and it is not generally harmful to the human body.

Step by step solution

01

Find the wavelength of the photon

To find the wavelength of the photon, we will use the formula: speed of light (c) = frequency (f) × wavelength (λ) We have the frequency (f) = 42.58 MHz = 42.58 × 10^6 Hz, and the speed of light (c) = 3.00 × 10^8 m/s. Rearranging the formula for wavelength, we get: wavelength (λ) = c / f Now plug in the values: λ = (3.00 × 10^8 m/s) / (42.58 × 10^6 Hz) Calculating the wavelength: λ ≈ 7.045 × 10^-3 m The wavelength of the photon is 7.045 × 10^-3 m, or 7.045 mm.
02

Calculate the energy of the photon

To find the energy, we will use the formula: energy (E) = Planck's constant (h) × frequency (f) Planck's constant (h) approximately equals 6.63 × 10^-34 Js, and the frequency (f) is 42.58 × 10^6 Hz. Now plug in the values: E = (6.63 × 10^-34 Js) × (42.58 × 10^6 Hz) Calculating the energy: E ≈ 2.823 × 10^-26 J The energy of this photon is 2.823 × 10^-26 J.
03

Identify the region of the electromagnetic spectrum

The wavelength of the photon is 7.045 × 10^-3 m, or 7.045 mm. Comparing this wavelength to the electromagnetic spectrum, we can find the region it belongs to. Since the value for the wavelength is between 1 mm and 100 mm, it is located in the radio wave region of the electromagnetic spectrum.
04

Determine the potential harm to human body

The photon is in the radio wave region of the electromagnetic spectrum, which is a non-ionizing form of radiation. This type of radiation does not have enough energy to remove tightly bound electrons from atoms, and thus, generally does not have enough energy to damage DNA or cause harmful effects to humans. In typical NMR experiments, the exposure to radio waves is minimal, and the intensity of the radiation is low, so we consider this photon emission to be safe for the human body. In summary: - Wavelength: 7.045 × 10^-3 m (7.045 mm) - Energy: 2.823 × 10^-26 J - Region of the spectrum: Radio waves - Harmful to the human body: No

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