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Chapter 22: Problem 20

Magnetic resonance imaging (MRI) is a medical technique for producing pictures of the interior of the body. The patient is placed within a strong magnetic field. One safety concern is what would happen to the positively and negatively charged particles in the body fluids if an equipment failure caused the magnetic field to be shut off suddenly. An induced emf could cause these particles to flow, producing an electric current within the body. Suppose the largest surface of the body through which flux passes has an area of \(0.032 \mathrm{m}^{2}\) and a normal that is parallel to a magnetic field of \(1.5 \mathrm{T}\). Determine the smallest time period during which the field can be allowed to vanish if the magnitude of the average induced emf is to be kept less than \(0.010 \mathrm{V}\).

Short Answer

Expert verified
The smallest time period is 4.8 seconds.

Step by step solution

01

Understand the Problem Setup

We need to find the smallest time period during which the magnetic field can vanish to keep the induced emf less than the given value. We're provided the area of the body surface \(A = 0.032 \, \text{m}^2\), magnetic field strength \(B = 1.5 \, \text{T}\), and the maximum emf \(\varepsilon = 0.010 \, \text{V}\).
02

Use Faraday's Law of Induction

Faraday's Law of electromagnetic induction is given by \(\varepsilon = -\frac{d\Phi_B}{dt}\), where \(\Phi_B\) is the magnetic flux. The negative sign indicates Lenz's Law, but here we are concerned with the magnitude, so \(|\varepsilon| = \left|\frac{d\Phi_B}{dt}\right|\).
03

Calculate Magnetic Flux

The magnetic flux \(\Phi_B\) through a surface is \(\Phi_B = B \cdot A \cdot \cos(\theta)\).Since the normal is parallel to the magnetic field, \(\theta = 0\) which implies \(\cos(0) = 1\).Therefore, \(\Phi_B = B \cdot A = 1.5 \, \text{T} \cdot 0.032 \, \text{m}^2 = 0.048 \, \text{T}\cdot\text{m}^2\).
04

Rearrange Faraday's Law to Solve for Time

We know that \(|\varepsilon| = \left|\frac{\Delta \Phi_B}{\Delta t}\right|\) and we want the field to change from initial flux \(\Phi_B=0.048 \, \text{T}\cdot\text{m}^2\) to \(0\). Set \(|\varepsilon| = 0.010 \, \text{V}\):\[0.010 = \frac{0.048}{\Delta t}\] Solving for \(\Delta t\) gives:\[\Delta t = \frac{0.048}{0.010} = 4.8 \, \text{s}\].

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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, commonly known as MRI, is a non-invasive medical imaging technique used to visualize detailed internal structures. It works by using strong magnetic fields and radio waves to generate images of the organs in the body. This is beneficial in diagnosing a variety of medical conditions.

In an MRI machine, a person is placed inside a strong magnetic field, often around 1.5 Tesla or higher. This field aligns the hydrogen nuclei in water molecules throughout the body. When a radiofrequency current is pulsed through the patient, the protons are knocked out of alignment, releasing energy that the MRI sensors detect as they return to their original alignment.

MRI is praised for its safety, as it doesn't use ionizing radiation like X-rays or CT scans. However, the strong magnetic fields involved bring certain safety considerations, notably related to Faraday's Law of electromagnetic induction, where a sudden change in the magnetic environment could potentially induce currents in body tissues.
Magnetic Flux
Magnetic flux is a measure of the magnetic field passing through a given surface. It is represented by the symbol \( \Phi_B \) and is calculated by the product of the magnetic field strength \( B \), the area \( A \) through which the field lines pass, and the cosine of the angle between the field lines and the normal (perpendicular) to the surface.

\[ \Phi_B = B \cdot A \cdot \cos(\theta) \]

In simpler terms, magnetic flux provides a way to describe how much of the magnetic field is "flowing" through a certain area. If the field is perpendicular to the surface, \( \cos(0) = 1 \), and the maximum flux is transmitted.

In the context of MRI and the given problem, the magnetic flux becomes crucial when determining how changes in the magnetic field can induce an electromotive force (emf). When thinking about the safety in MRI, understanding how the flux behaves and changes can help in designing safer operation protocols.
Electromagnetic Induction
Electromagnetic induction is a fundamental phenomenon where an electric voltage (emf) is generated by a change in magnetic flux. This principle was famously described by Faraday's Law, which states that the induced emf is directly proportional to the rate of change of magnetic flux.

\[ \varepsilon = -\frac{d\Phi_B}{dt} \]

This concept is pervasive in many technologies, such as electric generators, transformers, and indeed MRI machines. In the exercise, when the magnetic field in an MRI changes, it can induce a current due to electromagnetic induction. This potential for induced currents in the body underpins some safety concerns related to MRI usage.

Faraday's Law helps us calculate the induced emf and allows us to reason about the time over which abrupt changes in magnetic fields should occur to avoid adverse effects, such as unnecessary currents in the body. Understanding and applying electromagnetic induction ensures that technologies like MRI remain both effective and safe.

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Most popular questions from this chapter

A conducting coil of 1850 turns is connected to a galvanometer, and the total resistance of the circuit is \(45.0 \Omega\). The area of each turn is \(4.70 \times\) \(10^{-4} \mathrm{m}^{2} .\) This coil is moved from a region where the magnetic field is zero into a region where it is nonzero, the normal to the coil being kept parallel to the magnetic field. The amount of charge that is induced to flow around the circuit is measured to be \(8.87 \times 10^{-3} \mathrm{C} .\) Find the magnitude of the magnetic field.

Mutual induction can be used as the basis for a metal detector. A typical setup uses two large coils that are parallel to each other and have a common axis. Because of mutual induction, the ac generator connected to the primary coil causes an emf of \(0.46 \mathrm{V}\) to be induced in the secondary coil. When someone without metal objects walks through the coils, the mutual inductance and, thus, the induced emf do not change much. But when a person carrying a handgun walks through, the mutual inductance increases. The change in emf can be used to trigger an alarm. If the mutual inductance increases by a factor of three, find the new value of the induced emf.

A standard door into a house rotates about a vertical axis through one side, as defined by the door's hinges. A uniform magnetic field is parallel to the ground and perpendicular to this axis. Through what angle must the door rotate so that the magnetic flux that passes through it decreases from its maximum value to one-third of its maximum value?

A flat coil of wire has an area \(A, N\) turns, and a resistance \(R .\) It is situated in a magnetic field, such that the normal to the coil is parallel to the magnetic field. The coil is then rotated through an angle of \(90^{\circ},\) so that the normal becomes perpendicular to the magnetic field. The coil has an area of \(1.5 \times 10^{-3} \mathrm{m}^{2}, 50\) turns, and a resistance of \(140 \Omega .\) During the time while it is rotating, a charge of \(8.5 \times 10^{-5} \mathrm{C}\) flows in the coil. What is the magnitude of the magnetic field?

Consult Multiple-Concept Example 11 for background material relating to this problem. A small rubber wheel on the shaft of a bicycle generator presses against the bike tire and turns the coil of the generator at an angular speed that is 38 times as great as the angular speed of the tire itself. Each tire has a radius of \(0.300 \mathrm{m}\). The coil consists of 125 turns, has an area of \(3.86 \times 10^{-3} \mathrm{m}^{2},\) and rotates in a \(0.0900-\mathrm{T}\) magnetic field. The bicycle starts from rest and has an acceleration of \(+0.550 \mathrm{m} / \mathrm{s}^{2} .\) What is the peak emf produced by the generator at the end of 5.10 s?
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