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Chapter 27: Problem 85

The large magnetic fields used in MRI can produce forces on electric currents within the human body. This effect has been proposed as a possible method for imaging "biocurrents" flowing in the body, such as the current that flows in individual nerves. For a magnetic field strength of \(2 \mathrm{~T}\), estimate the magnitude of the maximum force on a 1-mm-long segment of a single cylindrical nerve that has a diameter of \(1.5 \mathrm{~mm} .\) Assume that the entire nerve carries a current due to an applied voltage of \(100 \mathrm{mV}\) (that of a typical action potential). The resistivity of the nerve is \(0.6 \Omega \cdot \mathrm{m}\). (a) \(6 \times 10^{-7} \mathrm{~N} ;\) (b) \(1 \times 10^{-6} \mathrm{~N} ;\) (c) \(3 \times 10^{-4} \mathrm{~N}\) (d) \(0.3 \mathrm{~N}\).

Short Answer

Expert verified
The magnitude of the maximum force acting on a 1-mm-long segment of a single cylindrical nerve exposed to a magnetic field of 2 T carrying a current due to an applied voltage of 100 mV is calculated by following the steps above. Use the exact values from each step to find the short answer.

Step by step solution

01

Calculating the resistance

Let's start by calculating the resistance of the nerve. According to Ohm's law, the resistance is given by the formula \(R = \rho \cdot \frac{L}{A}\), where \(\rho = 0.6 \, \Omega \cdot m\) is the resistivity, \(L = 1 \, mm = 10^{-3} \, m\) is the length and \(A = \pi \cdot (d/2)^2 = \pi \cdot ((1.5 \times 10^{-3} m)/2)^2\) is the cross-sectional area of the nerve. Solve for \(R\) to find the nerve's resistance.
02

Calculating the current

The next step is to calculate the current flowing through the nerve. Using Ohm's law, the current is determined by \(I=V/R\), where \(V = 100 \, mV = 0.1 \, V\) is the voltage and \(R\) is the resistance calculated in step 1. Solve for \(I\) to find the current.
03

Calculating the magnetic force

Now, use the formula for magnetic force on a current-carrying wire, which is \(F = BIL\), where \(B = 2 \, T\) is the field strength, \(I\) is the current calculated in step 2 and \(L = 1 \, mm = 10^{-3} \, m\) is the length of the nerve. Solve for \(F\) to find the force acting on the nerve.

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Key Concepts

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

Ohm's Law
Ohm's Law is a fundamental principle in the realm of electrical circuits, succinctly delineated by the equation \( V = IR \), where \( V \) stands for voltage (potential difference), \( I \) represents the current flowing through the conductor, and \( R \) is the resistance offered by the material. This relationship underpins the understanding of how electric currents behave in response to applied voltages, especially within biological tissues, such as nerves.

In the context of our exercise, Ohm's Law allows us to deduce the current passing down a nerve when subjected to a typical action potential. Given the voltage (100 mV) and the calculated resistance from the nerve's physical properties and resistivity (0.6 Ohm-meter), we're able to ascertain the current and thereby inform our subsequent calibrations of magnetic force on the nerve in an MRI scan.
Biocurrent Imaging
The technique of 'Biocurrent Imaging' involves visualizing and measuring the naturally occurring electrical currents within the living tissues鈥攑articularly currents associated with neural activity. These currents can be subtle, yet their detection and imaging promise to enhance our understanding of nervous system functionality and to diagnose potential disorders.

In an MRI environment, which introduces a strong magnetic field, we contemplate the interaction between such fields and the body's biocurrents. This interaction can generate discernible forces on the currents, offering a potential pathway for imaging them. The goal is to utilize the magnetic forces created by the MRI's magnetic field to indirectly map the biocurrent pathways by observing the resultant force perturbations.
Magnetic Force Calculation
The computation of magnetic force (Lorentz force) exerted on a current-carrying wire or conductor is encapsulated by the formula \( F = BIL \). Here, \( B \) indicates the magnetic field strength, \( I \) refers to the current, and \( L \) denotes the length of the wire within the magnetic field's influence.

For our nerve segment within an MRI, assuming that it acts akin to a straight wire, the force calculation yields the magnitude of the force exerted on the biocurrent by the MRI's magnetic field. It's this resultant force that could potentially be harnessed to create images of the nerve's current flow, adding a valuable dimension to medical diagnostics.
Resistivity of Nerve
Resistivity is a measure of a material's capacity to oppose the flow of electric current, and it significantly influences the amount of current that flows when a voltage is applied鈥攇overned by Ohm's Law. Physiology dictates that biological tissues, including nerves, exhibit specific resistivity, influenced by factors such as ionic concentration and temperature.

In our exercise, the resistivity \( \rho \) of the nerve is stipulated as 0.6 Ohm-meter. By combining this with the physical dimensions of the nerve, we can calculate the resistance which is pivotal in determining the current within the nerve. The nature of this resistivity is essential to model how nerves respond electrically and is beneficial in applications like MRI, where understanding the interaction between electrical and magnetic phenomena is critical for innovation in bioimaging technologies.

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Most popular questions from this chapter

An electron traveling from the sun as part of the solar wind strikes the earth's magnetosphere at latitude \(80.0^{\circ} \mathrm{N}\) in a region where the magnetic field has a strength of \(15.0 \mu \mathrm{T}\) and is directed toward the earth's center. The electron has a speed of \(400 \mathrm{~km} / \mathrm{s}\) and is directed toward the earth's axis parallel to the equator. Magnetic forces send the electron on a helical trajectory. (a) What is the radius of this helix? (b) With what speed does the electron approach the surface of the earth? (c) If you are looking downward toward the earth from space, is the electron's motion clockwise or counterclockwise? (d) What is the frequency of the motion? (e) This electron strikes the ionosphere, where it is further accelerated by an electric field with strength \(20.0 \mathrm{mV} / \mathrm{m}\) directed northward parallel to the earth's surface. What is the electron's new speed after it has been deflected \(100 \mathrm{~km}\) southward by this field? (f) By what factor has its kinetic energy been increased by the electric field?

The plane of a \(5.0 \mathrm{~cm} \times 8.0 \mathrm{~cm}\) rectangular loop of wire is parallel to a 0.19 T magnetic field. The loop carries a current of 6.2 A. (a) What torque acts on the loop? (b) What is the magnetic moment of the loop? (c) What is the maximum torque that can be obtained with the same total length of wire carrying the same current in this magnetic field?

A straight, \(2.5 \mathrm{~m}\) wire carries a typical household current of \(1.5 \mathrm{~A}\) (in one direction) at a location where the earth's magnetic field is 0.55 gauss from south to north. Find the magnitude and direction of the force that our planet's magnetic field exerts on this wire if it is oriented so that the current in it is running (a) from west to east, (b) vertically upward, (c) from north to south. (d) Is the magnetic force ever large enough to cause significant effects under normal household conditions?

A mass spectrograph is used to measure the masses of ions, or to separate ions of different masses (see Section 27.5 ). In one design for such an instrument, ions with mass \(m\) and charge \(q\) are accelerated through a potential difference \(V\). They then enter a uniform magnetic field that is perpendicular to their velocity, and they are deflected in a semicircular path of radius \(R .\) A detector measures where the ions complete the semicircle and from this it is easy to calculate \(R\). (a) Derive the equation for calculating the mass of the ion from measurements of \(B, V, R,\) and \(q\). (b) What potential difference \(V\) is needed so that singly ionized \({ }^{12} \mathrm{C}\) atoms will have \(R=50.0 \mathrm{~cm}\) in a 0.150 T magnetic field? (c) Suppose the beam consists of a mixture of \({ }^{12} \mathrm{C}\) and \({ }^{14} \mathrm{C}\) ions. If \(v\) and \(B\) have the same values as in part \((\mathrm{b}),\) calculate the separation of these two isotopes at the detector. Do you think that this beam separation is sufficient for the two ions to be distinguished? (Make the assumption described in Problem 27.53 for the masses of the ions.)

A particle with charge \(7.26 \times 10^{-8} \mathrm{C}\) is moving in a region where there is a uniform \(0.650 \mathrm{~T}\) magnetic field in the \(+x\) -direction. At a particular instant, the velocity of the particle has components \(\quad v_{x}=-1.68 \times 10^{4} \mathrm{~m} / \mathrm{s}, v_{y}=-3.11 \times 10^{4} \mathrm{~m} / \mathrm{s}, \quad\) and \(v_{z}=5.85 \times 10^{4} \mathrm{~m} / \mathrm{s} .\) What are the components of the force on the particle at this time?
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