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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 (frequency \(=1.23 \times 10^{20} \mathrm{Hz}\)).

Short Answer

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

Step by step solution

01

Understanding the Relationship Between Frequency and Wavelength

The relationship between the speed of light \( c \), frequency \( f \), and wavelength \( \lambda \) of an electromagnetic wave is given by the formula: \[ c = f \times \lambda \]where \( c \) is the speed of light \( \approx 3 \times 10^8 \text{ m/s} \). To find the wavelength from the frequency, rearrange the formula to:\[ \lambda = \frac{c}{f} \]
02

Calculate the Wavelength for MRI

Using the formula from Step 1: \[ \lambda_{\text{MRI}} = \frac{3 \times 10^8 \text{ m/s}}{6.38 \times 10^7 \text{ Hz}} \]Calculate \( \lambda_{\text{MRI}} \):\[ \lambda_{\text{MRI}} = 4.70 \text{ m} \]
03

Calculate the Wavelength for PET Scanning

Using the formula from Step 1: \[ \lambda_{\text{PET}} = \frac{3 \times 10^8 \text{ m/s}}{1.23 \times 10^{20} \text{ Hz}} \]Calculate \( \lambda_{\text{PET}} \):\[ \lambda_{\text{PET}} = 2.44 \times 10^{-12} \text{ m} \]
04

Calculate the Ratio of MRI Wavelength to PET Scanning Wavelength

Now, find the ratio of the two wavelengths: \[ \text{Ratio} = \frac{\lambda_{\text{MRI}}}{\lambda_{\text{PET}}} = \frac{4.70 \text{ m}}{2.44 \times 10^{-12} \text{ m}} \]Calculate the ratio:\[ \text{Ratio} \approx 1.93 \times 10^{12} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Resonance Imaging (MRI)
Magnetic Resonance Imaging, or MRI, is a medical imaging technique that captures detailed images of the interior of the body. It is primarily used for diagnosing a variety of conditions such as tumors, brain disorders, and joint injuries. MRI utilizes powerful magnets and radio waves to generate cross-sectional images without using ionizing radiation like X-rays.

In MRI, the human body is placed within a strong magnetic field, aligning hydrogen protons within the body temporarily. When the magnetic field is adjusted, these protons emit radio waves as they return to their normal state. These signals are captured to create detailed images.

MRI relies on low-frequency electromagnetic waves with a frequency around the range of tens of megahertz, typically around 6.38 脳 10鈦 Hz. The corresponding wavelength can be calculated using the speed of light formula, confirming the safe use and efficacy of this technology in medical diagnostics.
Positron Emission Tomography (PET)
Positron Emission Tomography, or PET, is an advanced imaging technique that examines the metabolic processes within the body. It is particularly useful in oncology, neurology, and cardiology, helping to detect cancer, heart diseases, and brain disorders.

PET works by injecting a small amount of radioactive material into the body, which emits positrons. When these positrons encounter electrons, they annihilate, producing gamma rays that are detected to form an image. The energy and location of these gamma rays provide insights into the body's metabolic activity.

PET scanning involves high-frequency electromagnetic waves, with frequencies around 1.23 脳 10虏鈦 Hz. This is significantly higher than those used in MRI, leading to much shorter wavelengths. The PET technology ensures detailed metabolic activity visualization, offering crucial information for treatment planning.
Wavelength and Frequency Relationship
The relationship between wavelength and frequency is a fundamental concept in understanding electromagnetic waves. This relationship is governed by the formula: \[ c = f \times \lambda \]where \( c \) is the speed of light (approximately 3 脳 10鈦 m/s), \( f \) is the frequency, and \( \lambda \) is the wavelength.

To find the wavelength from a given frequency, the formula rearranges to:\[ \lambda = \frac{c}{f} \]This highlights how frequency and wavelength are inversely proportional; as one increases, the other decreases.
  • For MRI, with a frequency of 6.38 脳 10鈦 Hz, the wavelength is relatively long, around 4.70 meters.
  • For PET scans, with a frequency of 1.23 脳 10虏鈦 Hz, the wavelength is very short, approximately 2.44 脳 10鈦宦孤 meters.
The different wavelengths and frequencies are purposely chosen for their specific applications in medical imaging, providing both safety and precision in capturing comprehensive images.

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Most popular questions from this chapter

A truck driver is broadcasting at a frequency of 26.965 MHz with a CB (citizen's band) radio. Determine the wavelength of the electromagnetic wave being used. The speed of light is \(c=2.9979 \times 10^{8} \mathrm{m} / \mathrm{s}\).

An electromagnetic wave strikes a \(1.30-\mathrm{cm}^{2}\) section of wall perpendicularly. The rms value of the wave's magnetic field is determined to be \(6.80 \times 10^{-4}\) T. How long does it take for the wave to deliver \(1850 \mathrm{J}\) of energy to the wall?

A politician holds a press conference that is televised live. The sound picked up by the microphone of a TV news network is broadcast via electromagnetic waves and heard by a television viewer. This vicwer is seated \(2.3 \mathrm{m}\) from his television set. A reporter at the press conference is located \(4.1 \mathrm{m}\) from the politician, and the sound of the words travels directly from the celebrity's mouth, through the air, and into the reporter's ears. The reporter hears the words exactly at the same instant that the television viewer hears them. Using a value of \(343 \mathrm{m} / \mathrm{s}\) for the speed of sound, determine the maximum distance between the television set and the politician. Ignore the small distance between the politician and the microphone. In addition, assume that the only delay between what the microphone picks up and the sound being emitted by the television set is that due to the travel time of the electromagnetic waves used by the network.

Two radio waves are used in the operation of a cellular telephone. To receive a call, the phone detects the wave emitted at one frequency by the transmitter station or base unit. To send your message to the base unit, your phone emits its own wave at a different frequency. The difference between these two frequencies is fixed for all channels of cell phone operation. Suppose that the wavelength of the wave emitted by the base unit is \(0.34339 \mathrm{m}\) and the wavelength of the wave emitted by the phone is \(0.36205 \mathrm{m} .\) Using a value of \(2.9979 \times 10^{8} \mathrm{m} / \mathrm{s}\) for the speed of light, determine the difference between the two frequencies used in the operation of a cell phone.

In a dentist's office an X-ray of a tooth is taken using X-rays that have a frequency of \(6.05 \times 10^{18} \mathrm{Hz}\). What is the wavelength in vacuum of these X-rays?

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