Problem 22 - University Physics with Modern Physics

Question

Currents in dc transmission lines can be 100 A or higher. Some people are concerned that the electromagnetic fields from such lines near their homes could pose health dangers. For a line that has current 150 A and a height of 8.0 m above the ground, what magnetic field does the line produce at ground level? Express your answer in teslas and as a percentage of the earth's magnetic field, which is 0.50 G. Is this value cause for worry?

Step-by-Step Solution

Answer

The magnetic field is 3.75 µT, 7500% of Earth's field. This value doesn't instantly imply health hazards but requires further context analysis based on regulations.

Step 1: Understanding the Concept

This problem can be solved using the formula for the magnetic field around a long straight conductor, which is given by Ampere's Law. The formula for the magnetic field at a distance \( r \) from a long, straight conductor carrying current \( I \) is \( B = \frac{\mu_0 I}{2\pi r} \), where \( \mu_0 \) is the permeability of free space \( (4\pi \times 10^{-7} \text{ T m/A}) \).

Step 2: Substituting the Given Values

We have \( I = 150 \text{ A} \) and \( r = 8.0 \text{ m} \) (the height of the power line above the ground). Substitute these values into the formula: \[ B = \frac{(4\pi \times 10^{-7}) \times 150}{2\pi \times 8} \] Simplifying this expression gives the magnetic field at ground level.

Step 3: Simplifying the Equation

Perform the calculations:\[ B = \frac{4\pi \times 10^{-7} \times 150}{2\pi \times 8} = \frac{4 \times 150 \times 10^{-7}}{16} \] This further simplifies to:\[ B = \frac{600 \times 10^{-7}}{16} = 37.5 \times 10^{-7} \text{ T} \] So the magnetic field at ground level is \( 3.75 \times 10^{-6} \text{ T} \).

Step 4: Expressing the Result in Other Units

Convert this value into teslas and gauss. Note that \( 1 \text{ T} = 10^4 \text{ G} \), consequently:\[ 3.75 \times 10^{-6} \text{ T} \equiv 37.5 \text{ G} \]

Step 5: Calculating the Percentage of Earth's Magnetic Field

Earth's magnetic field is \( 0.50 \text{ G} \). Calculate the percentage of the magnetic field produced by the power line compared to Earth's magnetic field:\[ \frac{37.5 \text{ G}}{0.50 \text{ G}} \times 100\% = 7500\% \]

Step 6: Conclusion on Safety

The magnetic field from the power line is 7500% of the Earth's magnetic field. However, regulations and health studies generally focus on long-term exposure and specific health effects, which often still debate safety thresholds and implications.

Key Concepts

magnetic field

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. It is invisible to the eye, but can be detected using tools like a compass or a magnetometer. In the context of a dc transmission line, this field is a result of the moving electric charges in the current.
The magnetic field lines form concentric circles around the wire when a current passes through it. This is fundamental, especially when considering how electrical currents affect their surroundings.

• The strength of the magnetic field generated by a current-carrying wire diminishes with increased distance from the wire.
• The direction of the field can be determined using the 'right-hand rule' – point your thumb in the direction of the current, and your fingers will curl in the direction of the magnetic field lines.
• In the case of dc transmission lines, these magnetic fields are often scrutinized for potential health effects, though scientific consensus remains inconclusive on direct links to health problems.

Ampere's Law

Ampere's Law is one of the fundamental principles in electromagnetism that relates magnetic fields to the electric currents that produce them. It is expressed through the formula:
\( B = \frac{\mu_0 I}{2\pi r} \), where:

• \( B \) is the magnetic field magnitude.
• \( \mu_0 \) is the permeability of free space, a constant value.
• \( I \) represents the current flowing through the conductor.
• \( r \) is the distance from the conductor.

Ampere's Law assumes that the conductor is infinitely long and straight, which simplifies the calculations of magnetic fields significantly.
This law helps in understanding how electrical currents interact with their environment, making it invaluable for designing circuits and in applications involving electromagnetic fields.Using Ampere's Law, the derived formula allows us to calculate the magnetic field produced by a dc transmission line, as illustrated above. It efficiently connects the quantitative aspects of electrical engineering to practical scenarios encountered in everyday life, including analyzing the impact of power lines.

permeability of free space

The permeability of free space, also known as the magnetic constant (\( \mu_0 \)), is a fundamental physical constant. It describes how a magnetic field can permeate through the vacuum of space and plays a crucial role in the relationships described by Ampere's Law.
The value of \( \mu_0 \) is fixed as \( 4\pi \times 10^{-7} \text{ T m/A} \), which gives us a standard measure for magnetic fields when used in equations.

• Being a universal constant, it simplifies the calculation of magnetic fields.
• It provides a basis for comparing magnetic properties of different media, as alterations from this constant indicate relative permeability of other materials.
• In practical terms, \( \mu_0 \) is central to electromagnetism and underpins how engineers create and understand electronic applications.

When we consider dc transmission lines, the permeability of free space helps to predict the strength of the magnetic field that might influence areas close to these lines. Understanding this constant is essential in both educational contexts and practical engineering disciplines.