Problem 83 - College Physics

Question

The effect of transmission lines. Two hikers are reading a compass under an overhead transmission line that is 5.50 \(\mathrm{m}\) above the ground and carries a current of 0.800 \(\mathrm{kA}\) in a horizontal direction from north to south. (a) Find the magnitude and direction of the magnetic field at a point on the ground directly under the transmission line. (b) One hiker suggests that they walk 50 \(\mathrm{m}\) away from the lines to avoid inaccurate compass readings due to the current. Considering that the earth's magnetic field is on the order of \(0.5 \times 10^{-4} \mathrm{T},\) is the current really a problem?

Step-by-Step Solution

Answer

The magnetic field under the line is \(2.91 \times 10^{-5} \, \mathrm{T}\) eastward. It's weaker than Earth's magnetic field, minimizing compass issues.

Step 1: Understanding the Problem

We need to find the magnetic field caused by a transmission line carrying a 0.800 kA current in a horizontal direction from north to south. The point of interest is directly beneath the line at a distance of 5.50 m. We will use Ampère's Law for a long straight conductor which describes the magnetic field around the wire.

Step 2: Apply Ampère's Law

Ampère's Law for a long straight wire gives the magnetic field at a distance \( r \) from the wire as \( B = \frac{\mu_0 I}{2 \pi r} \). Where \( \mu_0 \) (the permeability of free space) is \( 4\pi \times 10^{-7} \, \mathrm{T}\cdot\mathrm{m/A} \).

Step 3: Calculate the Magnetic Field Directly Under the Line

Substitute the given values: \( I = 0.800 \, \mathrm{kA} = 800 \, \mathrm{A} \) and \( r = 5.50 \, \mathrm{m} \). Calculate \( B \): \[ B = \frac{4\pi \times 10^{-7} \times 800}{2\pi \times 5.50} \approx 2.91 \times 10^{-5} \, \mathrm{T} \]

Step 4: Determine the Direction of the Magnetic Field

Using the right-hand rule, point your thumb in the direction of current flow (from north to south). Your fingers curl to point towards the east when below the wire, indicating that below the line, the magnetic field is directed eastward.

Step 5: Compare with the Earth's Magnetic Field

The magnetic field of the transmission line at the ground beneath is approximately \( 2.91 \times 10^{-5} \, \mathrm{T} \), whereas the Earth's magnetic field is \( 0.5 \times 10^{-4} \, \mathrm{T} = 5.0 \times 10^{-5} \, \mathrm{T} \). The field from the transmission line is smaller than Earth's magnetic field, so it doesn't cause significant deviation.

Step 6: Conclusion: Assess the Need to Walk Away

Since the transmission line's magnetic field is significantly weaker than Earth's magnetic field, the compass reading should remain fairly accurate even directly beneath the wire. Walking 50 meters away would reduce the influence even further, but it may not be necessary given the weaker field.

Key Concepts

Ampère's Law

Ampère's Law is a fundamental principle used to calculate the magnetic field generated by an electric current. It states that the magnetic field around a closed loop is proportional to the total current passing through the loop. This is especially useful for long straight conductors and helps us understand how magnetic fields behave around transmission lines.For a long, straight wire carrying a steady current, the magnetic field at a distance \( r \) from the wire is given by:\[ B = \frac{\mu_0 I}{2 \pi r} \]- \( B \) is the magnetic field strength.- \( \mu_0 \) is the permeability of free space.- \( I \) is the current.- \( r \) is the radial distance from the wire.This formula reflects how the magnetic field decreases with distance from the wire and is crucial for computing the field experienced directly under transmission lines, such as when hikers are trying to read a compass in these conditions.

permeability of free space

The permeability of free space, commonly denoted as \( \mu_0 \), is a fundamental physical constant that describes how magnetic fields are influenced by a vacuum. It is essential in equations that involve magnetic fields, particularly in Ampère's Law. Its value is \[ \mu_0 = 4\pi \times 10^{-7} \, \mathrm{T \cdot m/A} \]- \( \mu_0 \) determines how a magnetic field interacts with the medium it is in.- It acts as a scaling factor that influences the strength and behavior of the magnetic field generated by a current.Knowing \( \mu_0 \) helps in computing the magnetic fields around current-carrying conductors. For example, when calculating the magnetic field under a transmission line, \( \mu_0 \) ensures we can accurately gauge how strong that field will be and how it compares to other surrounding magnetic influences.

right-hand rule

The right-hand rule is a simple yet powerful mnemonic for determining the direction of the magnetic field around a current-carrying conductor. It provides a quick way to visualize the orientation of field lines. Here's how it works: - Extend your right hand. - Point your thumb in the direction of the current flow. - Let your fingers naturally curl around the conductor; they will point in the direction of the magnetic field lines. In the context of our original exercise, where the current runs from north to south, using the right-hand rule indicates that the magnetic field at a point below the wire will be directed towards the east. This intuitive tool is essential for predicting the effects of magnetic fields in real-world situations, such as determining how they might affect compass readings under power lines.