Problem 54 - University Physics with Modern Physics

Question

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?

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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

Key Concepts

Electromagnetic Spectrum

The electromagnetic spectrum is a range of all types of electromagnetic radiation. Radiation is energy that travels and spreads out as it goes. This spectrum is arranged by wavelength and frequency, providing a visual representation of the variety of electromagnetic waves present in our universe. Waves can vary from very short gamma rays, which have the highest frequencies, to long radio waves, which have the lowest.

Let's consider important points about the electromagnetic spectrum:

• Gamma rays have the shortest wavelength but the highest energy.
• Visible light is only a small portion of the spectrum.
• Radio waves occupy the end of the spectrum with lower frequencies and longer wavelengths.

Understanding where a particular wave lands on the spectrum helps determine its properties and potential uses.

Radio Waves

Radio waves are a type of electromagnetic radiation with wavelengths in the electromagnetic spectrum longer than infrared light. They are used for wireless communication because of their ability to travel long distances and through obstacles like walls.

Here's more about radio waves:

• They range from around 1 millimeter to 100 kilometers in wavelength.
• Radio waves are non-ionizing, meaning they don't carry enough energy to ionize atoms or molecules.
• Applications include broadcasting, satellite communication, and even in medical imaging technologies like MRI, where they play a crucial role in nuclear magnetic resonance (NMR).

Due to their low energy, radio waves are typically considered safe for humans, as they do not have the power to damage cells.

Photon Energy

Photon energy is the amount of energy carried by a single photon. This energy is directly proportional to the frequency of the corresponding electromagnetic wave and can be calculated using Planck’s equation. Higher frequency photons, like gamma rays, have more energy than lower frequency photons, like radio waves.

The formula for calculating photon energy is:

• \( E = h \times f \)
• Here, \( E \) is energy, \( h \) is Planck's constant \((6.63 \times 10^{-34} \text{Js})\), and \( f \) is the frequency.

For a photon in the radio wave region, the energy will be on the lower end, making it safer for everyday exposure. This computation is essential for understanding the energy potential of different electromagnetic waves.

Wavelength Calculation

The wavelength of a wave is the distance over which the wave's shape repeats. It can be calculated if the frequency and speed of the wave are known, utilizing the relationship between frequency, speed of light, and wavelength. This concept is especially useful in identifying where a wave fits in the electromagnetic spectrum.

The equation used for wavelength calculation is:

• \( \lambda = \frac{c}{f} \)
• Here, \( \lambda \) is the wavelength, \( c \) is the speed of light \((3.00 \times 10^{8} \text{ m/s})\), and \( f \) is the frequency.

Applying the above equation helps to deduce the wavelength of radio waves, affirming their suitability for various applications based on their position in the spectrum. In nuclear magnetic resonance, understanding the wavelength is crucial for the correct interpretation of the signals received during the procedure.