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Chapter 28: Problem 24

BIO EMF. Currents in de transmission lines can be 100 A or more. Some people have expressed concern that the electromagnetic fields (EMFs) from such lines near their homes could cause health dangers. For a line with current 150 \(\mathrm{A}\) and at a height of 8.0 \(\mathrm{m}\) above the ground, what magnetic field does the line produce at ground level? Express your answer in teslas and as a percent of the earth's magnetic field, which is 0.50 gauss. Does this seem to be cause for worry?

Short Answer

Expert verified
The magnetic field at ground level is 3.75 \(\times 10^{-4}\) T, which is 750% of Earth's magnetic field. This increase might cause concern due to health impacts.

Step by step solution

01

Understanding the Problem

We are given a current-carrying transmission line 8.0 meters above the ground, carrying a current of 150 A. We need to find the magnetic field at ground level and compare it to Earth's magnetic field, which is 0.50 gauss.
02

Using the Formula for Magnetic Field Due to a Long Straight Wire

The formula for the magnetic field \( B \) at a distance \( r \) from a long straight wire carrying current \( I \) is given by \[ B = \frac{\mu_0 I}{2\pi r}, \]where \( \mu_0 = 4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A}\) is the permeability of free space. Here, \( I = 150 \ \text{A} \) and \( r = 8.0 \ \text{m}.\)
03

Calculating the Magnetic Field at the Ground Level

Substitute the given values into the formula:\[ B = \frac{4\pi \times 10^{-7} \times 150}{2\pi \times 8.0} = \frac{2 \times 10^{-5} \cdot 150}{8.0} \]\[ B = \frac{3.0 \times 10^{-3}}{8.0} = 3.75 \times 10^{-4} \ \text{T}. \]
04

Converting Earth's Magnetic Field to Tesla

Earth's magnetic field is given in gauss: 0.50 gauss. We need to convert this to tesla:\[ 1 \ \text{gauss} = 10^{-4} \ \text{T}. \]Therefore, \[ 0.50 \ \text{gauss} = 0.50 \times 10^{-4} \ \text{T} = 5.0 \times 10^{-5} \ \text{T}. \]
05

Calculating the Percent of Earth's Magnetic Field

Find what percent the magnetic field at ground level is of Earth's magnetic field:\[ \text{Percent} = \left(\frac{3.75 \times 10^{-4}}{5.0 \times 10^{-5}}\right) \times 100\% \]\[ \text{Percent} = \left(\frac{3.75}{0.5}\right) \times 100\% \approx 750\%. \]
06

Evaluating the Causes for Worry

The magnetic field from the transmission line at ground level is approximately 750% of Earth's magnetic field. This significant increase compared to Earth's natural magnetic field might be a cause for concern for some people regarding health, depending on the scientific studies related to such exposure.

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Key Concepts

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

Magnetic Field Calculation
Calculating the magnetic field generated by a current-carrying wire is an important aspect of understanding how electromagnetic fields (EMFs) behave. When a wire carries an electric current, it generates a magnetic field around it. To find the magnetic field at a certain distance from a long, straight wire carrying current, we use the formula:
  • \[ B = \frac{\mu_0 I}{2\pi r} \]
where: - \( B \) is the magnetic field,- \( \mu_0 \) is the permeability of free space - \( I \) denotes the current in amperes, and - \( r \) represents the distance from the wire in meters.
In this scenario, the transmission line carries a current of 150 A and is positioned 8.0 meters above the ground. By applying these values, we find that the magnetic field at ground level is
  • \[ 3.75 \times 10^{-4} \ \text{T} \]
This calculated magnetic field helps compare the strength of EMFs from the line to other sources, such as Earth's magnetic field.
Permeability of Free Space
The permeability of free space, denoted as \( \mu_0 \), is a fundamental constant in electromagnetics. It plays a crucial role in calculating magnetic fields produced by electric currents.
  • \( \mu_0 \) is defined as \( 4\pi \times 10^{-7} \ \text{T} \cdot\text{m/A} \).
This constant appears in the formula for the magnetic field due to a current in a wire, determining how effectively a magnetic field is generated in a vacuum or air.
Understanding \( \mu_0 \) is vital as it links the electric current to the resultant magnetic field. The larger the value of \( \mu_0 \), the more potent the magnetic field produced by a given current will be. This constant, therefore, ensures accurate calculations across various applications, from basic academic exercises to complex engineering designs.
Impact on Health
The effect of electromagnetic fields on health has caught the attention of scientists and the public alike. When dealing with high current transmission lines, understanding the potential health implications of EMF exposure is essential.
  • In our example, the magnetic field at ground level is about 750% that of Earth's natural magnetic field.
Such a significant increase in magnetic field strength might raise concerns regarding prolonged exposure.
Research into the health impact of EMFs has yielded mixed findings. Some studies suggest there may be a link between high EMF exposure and health issues such as headaches or even more severe conditions. However, consensus in the scientific community is that more research is needed to definitively determine risk levels.
While guidelines exist to limit exposure, living near high-current lines might still lead some residents to worry. Awareness and ongoing research remain crucial in addressing these health impact concerns effectively.

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Most popular questions from this chapter

Two long, parallel wires are separated by a distance of 2.50 \(\mathrm{cm} .\) The force per unit length that each wire exerts on the other is \(4.00 \times 10^{-5} \mathrm{N} / \mathrm{m},\) and the wires repel each other. The current in one wire is 0.600 A. (a) What is the current in the second wire? (b) Are the two currents in the same direction or in opposite directions?

BIO Bacteria Navigation. Certain bacteria (such as Aquaspirillum magnetotacticum) tend to swim toward the earth's geographic north pole because they contain tiny particles, called magnetosomes, that are sensitive to a magnetic field. If a transmission line carrying 100 \(\mathrm{A}\) is laid underwater, at what range of distances would the magnetic field from this line be great enough to interfere with the migration of these bacteria? (Assume that a field less than 5 percent of the earth's field would have little effect on the bacteria. Take the earth's field to be \(5.0 \times 10^{-5} \mathrm{T}\) and ignore the effects of the seawater.)

A toroidal solenoid with 500 turns is wound on a ring with a mean radius of 2.90 \(\mathrm{cm} .\) Find the current in the winding that is required to set up a magnetic field of 0.350 T in the ring (a) if the ring is made of annealed iron \(\left(K_{\mathrm{m}}=1400\right)\) and \((\mathrm{b})\) if the ring is made of silicon steel \(\left(K_{\mathrm{m}}=5200\right)\)

A solenoid is designed to produce a magnetic field of 0.0270 T at its center. It has radius 1.40 \(\mathrm{cm}\) and length \(40.0 \mathrm{cm},\) and the wire can carry a maximum current of 12.0 A. (a) What minimum number of turns per unit length must the solenoid have? (b) What total length of wire is required?

A long, straight, solid cylinder, oriented with its axis in the \(z\) -direction, carries a current whose current density is \(\vec{J}\) . The current density, although symmetric about the cylinder axis, is not constant and varies according to the relationship $$ \begin{aligned} \vec{J} &=\left(\frac{b}{r}\right) e^{(r-a) / \delta} \hat{k} & \text { for } r \leq a \\\ &=0 \quad \quad \text { for } r \geq a \end{aligned} $$ where the radius of the cylinder is \(a=5.00 \mathrm{cm}, r\) is the radial distance from the cylinder axis, \(b\) is a constant equal to \(600 \mathrm{A} / \mathrm{m},\) and \(\delta\) is a constant equal to 2.50 \(\mathrm{cm} .\) (a) Let \(I_{0}\) be the total current passing through the entire cross section of the wire. Obtain an expression for \(I_{0}\) in terms of \(b, \delta,\) and \(a\) . Evaluate your expression to obtain a numerical value for \(I_{0}\) . (b) Using Ampere's law, derive an expression for the magnetic field \(\vec{\boldsymbol{B}}\) in the region \(r \geq a .\) Express your answer in terms of \(I_{0}\) rather than \(b\) . (c) Obtain an expression for the current \(I\) contained in a circular cross section of radius \(r \leq a\) and centered at the cylinder axis. Express your answer in terms of \(I_{0}\) rather than \(b .\) (d) Using Ampere's law, derive an expression for the magnetic field \(\vec{\boldsymbol{B}}\) in the region \(r \leq a .\) (e) Evaluate the magnitude of the magnetic field at \(r=\delta, r=a,\) and \(r=2 a .\)
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