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Chapter 28: Problem 35

BIO Currents in the Brain. The magnetic field around the head has been measured to be approximately \(3.0 \times 10^{-8}\) G. Although the currents that cause this field are quite complicated, we can get a rough estimate of their size by modeling them as a single circular current loop 16 \(\mathrm{cm}\) (the width of a typical head) in diameter. What is the current needed to produce such a field at the center of the loop?

Short Answer

Expert verified
The current needed is approximately \(3.82 \times 10^{-7} \ \text{A}\).

Step by step solution

01

Understanding the Formula

To find the current needed to produce the given magnetic field, we can use the formula for the magnetic field of a circular loop at its center: \[B = \frac{\mu_0 I}{2R}\], where \(B\) is the magnetic field, \(\mu_0\) is the permeability of free space \(4\pi \times 10^{-7} \ \text{T m/A}\), \(I\) is the current, and \(R\) is the radius of the loop.
02

Converting Units and Values

The given magnetic field is \(3.0 \times 10^{-8} \ \text{G}\). We need to convert this to Tesla because the standard SI unit for magnetic fields in our formula is Tesla. Since \(1 \ \text{G} = 10^{-4} \ \text{T}\), the field in Tesla is \(3.0 \times 10^{-12} \ \text{T}\). Also, the diameter of the loop is 16 cm, so the radius \(R\) is \(0.08 \ \text{m}\) (since \(1 \ \text{cm} = 0.01 \ \text{m}\)).
03

Rearranging the Formula

We need to solve for the current \(I\). Rearrange the equation \(B = \frac{\mu_0 I}{2R}\) to find \(I\): \[I = \frac{2RB}{\mu_0}\].
04

Substituting the Values

Substitute the known values into the formula: \[I = \frac{2 \times 0.08 \ \text{m} \times 3.0 \times 10^{-12} \ \text{T}}{4\pi \times 10^{-7} \ \text{T m/A}}\].
05

Calculating the Current

Perform the calculations: \[I = \frac{4.8 \times 10^{-13}}{4\pi \times 10^{-7}}\]. Compute the denominator \(4\pi \times 10^{-7} \approx 1.25664 \times 10^{-6}\). Thus: \[I \approx \frac{4.8 \times 10^{-13}}{1.25664 \times 10^{-6}} \approx 3.82 \times 10^{-7} \ \text{A}\].

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Key Concepts

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

Magnetic Field Measurement
The measurement of magnetic fields, especially in the context of the human brain, is a fascinating subject. Some specialized equipment is required to detect such delicate fields. One of the most sensitive techniques used for this purpose is magnetoencephalography (MEG). MEG measures the magnetic fields produced by neural activity. What makes these measurements crucial is that they can reveal the brain's electrical activity in real-time.

The measurement of the magnetic field around the head, like in the original exercise, typically involves fields as small as 10 picoteslas. To put this into perspective, it is billions of times weaker than the Earth's magnetic field. Despite this challenge, the information gained through magnetic field measurements helps in understanding brain functions and diagnosing neurological disorders. This insight makes it a significant tool in both research and clinical settings.
Biological Currents
The concept of biological currents refers to the electrical currents produced by ion movements across cell membranes, such as those in neurons. In the brain, these currents are responsible for the propagation of electrical signals that govern everything from muscle contraction to thought processes.

Biological currents generate magnetic fields, which are what we measure when examining brain activity. Even though the currents are quite complex due to the myriad of neural connections, simplifying them to models like a circular current loop can help in estimating their effects. In the context of the original exercise, this simplification allowed us to calculate the current creating a specific magnetic field around the head. Understanding these currents and their magnetic fields enhances our grasp of how the human brain functions at a fundamental level.
Circular Current Loop
Modeling biological currents as a circular current loop is a simplified yet powerful way to understand how they generate magnetic fields in the brain. This model assumes the current is evenly distributed along a loop. Such a simplification is useful for computations and gaining insights into the behavior of magnetic fields.

In the context of our original problem, we calculate the magnetic field using the formula \(B = \frac{\mu_0 I}{2R}\), where \(B\) is the magnetic field at the center, \(\mu_0\) is the magnetic constant, \(I\) is the loop current, and \(R\) is the loop's radius. This approach allows us to estimate the current creating a given magnetic field strength by rearranging the formula to solve for \(I\). Such models, while not covering the complexity of the actual biological networks, provide a foundational understanding that can be expanded upon with more complex models and empirical data.

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Most popular questions from this chapter

(a) How large a current would a very long, straight wire have to carry so that the magnetic field 2.00 \(\mathrm{cm}\) from the wire is equal to 1.00 G (comparable to the earth's northward-pointing magnetic field)? (b) If the wire is horizontal with the current running from east to west, at what locations would the magnetic field of the wire point in the same direction as the horizontal component of the earth's magnetic field? (c) Repeat part (b) except the wire is vertical with the current going upward.

Fields Within the Atom. In the Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius \(5.3 \times 10^{-11} \mathrm{m}\) with a speed of \(2.2 \times 10^{6} \mathrm{m} / \mathrm{s} .\) If we are viewing the atom in such a way that the electron's orbit is in the plane of the paper with the electron moving clockwise, find the magnitude and direction of the electric and magnetic fields that the electron produces at the location of the nucleus (treated as a point).

A short current element \(d \vec{l}=(0.500 \mathrm{mm}) \hat{\jmath}\) carries a current of 8.20 \(\mathrm{A}\) in the same direction as \(d \vec{l} .\) Point \(P\) is located at \(\vec{r}=(-0.730 \mathrm{m}) \hat{\imath}+(0.390 \mathrm{m}) \hat{k} .\) Use unit vectors to express the magnetic field at \(P\) produced by this current element.

Two concentric circular loops of wire lie on a tabletop, one inside the other. The inner wire has a diameter of 20.0 \(\mathrm{cm}\) and carries a clockwise current of \(12.0 \mathrm{A},\) as viewed from above, and the outer wire has a diameter of 30.0 \(\mathrm{cm} .\) What must be the magnitude and direction (as viewed from above) of the current in the outer wire so that the net magnetic field due to this combination of wires is zero at the common center of the wires?

The Magnetic Field from a Lightning Bolt. Lightning bolts can carry currents up to approximately 20 kA. We can model such a current as the equivalent of a very long, straight wire. (a) If you were unfortunate enough to be 5.0 m away from such a lightning bolt, how large a magnetic field would you experience? (b) How does this field compare to one you would experience by being 5.0 \(\mathrm{cm}\) from a long, straight household current of 10 A ?
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