Problem 10 - Physics

Question

Magnetic resonance imaging, or MRI (see Section 21.7 ), and positron emission tomography, or PET scanning (see Section 32.6), are two medical diagnostic techniques. Both employ electromagnetic waves. For these waves, find the ratio of the MRI wavelength (frequency \(=6.38 \times 10^{7} \mathrm{~Hz}\) ) to the PET scanning wavelength \(\left(\right.\) frequency \(\left.=1.23 \times 10^{20} \mathrm{~Hz}\right)\)

Step-by-Step Solution

Answer

The ratio of the MRI wavelength to the PET scanning wavelength is approximately \(1.93 \times 10^{12}\).

Step 1: Understand the relationship between frequency, speed, and wavelength

The relationship between the speed of light (\(c\)) in a vacuum, frequency (\(f\)), and wavelength (\(\lambda\)) is given by the formula: \[ c = \lambda \times f \]where \(c = 3.00 \times 10^8 \text{ m/s}\). To find the wavelength, you can rearrange the formula to: \[ \lambda = \frac{c}{f} \].

Step 2: Calculate the MRI wavelength

Using the formula \( \lambda = \frac{c}{f} \), substitute \( c = 3.00 \times 10^{8} \text{ m/s}\) and the frequency for MRI \( f = 6.38 \times 10^{7} \text{ Hz} \):\[\lambda_{\text{MRI}} = \frac{3.00 \times 10^{8}}{6.38 \times 10^{7}} = 4.70 \text{ m}\]

Step 3: Calculate the PET scanning wavelength

Similarly, use the formula \( \lambda = \frac{c}{f} \) for the PET scanning:Substitute \( c = 3.00 \times 10^{8} \text{ m/s} \) and the PET frequency \( f = 1.23 \times 10^{20} \text{ Hz} \):\[\lambda_{\text{PET}} = \frac{3.00 \times 10^{8}}{1.23 \times 10^{20}} \approx 2.44 \times 10^{-12} \text{ m}\]

Step 4: Find the ratio of MRI wavelength to PET scanning wavelength

Now, calculate the ratio of the MRI wavelength to the PET scanning wavelength, which is:\[\text{Ratio} = \frac{\lambda_{\text{MRI}}}{\lambda_{\text{PET}}} = \frac{4.70}{2.44 \times 10^{-12}} = 1.93 \times 10^{12}\]

Key Concepts

Magnetic Resonance Imaging (MRI)

Magnetic Resonance Imaging, or MRI, is a cutting-edge technique that uses electromagnetic waves to create detailed images of organs and structures inside the body.

This non-invasive method relies on the interaction of magnetic fields and radio waves to map the distribution of hydrogen atoms in the body. MRI appearances are based on the hydrogen atoms, which are abundant in tissues like water and fats.

• **Safety and Comfort**: MRI doesn't use ionizing radiation, making it safer for repeated use when monitoring conditions over time.
• **Versatility**: It's versatile and can be used for imaging the brain, spinal cord, and musculoskeletal structures.

These attributes make MRI a remarkable tool for diagnosing a variety of conditions, such as tumors and injuries. Its ability to differentiate between different types of soft tissue is unparalleled.

Positron Emission Tomography (PET)

Positron Emission Tomography, or PET, is a fascinating technique that provides detailed images of functional processes in the body.

It works by using small amounts of radioactive materials known as radiotracers, a special camera, and a computer to evaluate organ and tissue functions. The radiotracers collect in areas of the body with high levels of chemical activity, which often correspond to disease areas.

• **Functional Imaging**: Unlike MRI, which is more anatomical, PET focuses on the metabolic activity of cells.
• **Applications**: It's especially useful in diagnosing cancers, neurological conditions, and heart diseases.

By providing insights into the body’s metabolic state, PET scanning excels in early detection of illnesses and monitoring of treatment progress.

Wavelength and Frequency Relationship

Understanding the relationship between wavelength and frequency is key to grasping how electromagnetic waves work.

Electromagnetic waves, whether in MRI or PET technologies, are characterized by their speed, which is constant in a vacuum at around \(3.00 \times 10^8\) m/s. The connection between speed (\(c\)), frequency (\(f\)), and wavelength (\(\lambda\)) is given by the formula: \[ c = \lambda \times f \].

• **Wavelength Calculation**: Rearranging gives \(\lambda = \frac{c}{f}\), allowing the calculation of wavelength when the frequency is known.
• **Inverse Relationship**: The frequency and wavelength are inversely proportional—higher frequencies mean shorter wavelengths and vice versa.

In the exercise, this relationship helps calculate the ratio of MRI to PET wavelengths, illustrating how varying frequencies affect wave propagation. This fundamental principle is crucial for further understanding and analyzing other wave-based technologies.