Problem 20 - University Physics with Modern Physics

Question

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 800 \(\mathrm{A}\) in a horizontal direction from north to south. (a) Find the magnitude and direction of the magnetic field at a point on the ground directiy under the conductor. (b) One hiker suggests they walk on another 50 \(\mathrm{m}\) to avoid inaccurate compass readings caused by the current. Considering that the magnitude of the earth's field is of the order of \(0.5 \times 10^{-4} \mathrm{T},\) is the current really a problem?

Step-by-Step Solution

Answer

The magnetic field from the line is 2.91 x 10^-5 T, weaker than the Earth's field (5 x 10^-5 T), so it should not significantly affect the compass.

Step 1: Understand the Problem

We need to find the magnetic field intensity due to the current in the transmission line above the hikers and decide if it affects the compass readings. We are given the line carries 800 A of current and is 5.50 m above the ground.

Step 2: Use Biot-Savart Law for Magnetic Field

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

Step 3: Calculate Magnetic Field Under the Conductor

Substitute the given values into the formula: \( I = 800 \ A \) and \( r = 5.50 \ m \). Hence, \( B = \frac{4\pi \times 10^{-7} \times 800}{2 \pi \times 5.50} \ = \frac{3200 \times 10^{-7}}{11} \ = 2.91 \times 10^{-5} \ T \).

Step 4: Compare with Earth's Magnetic Field

The Earth’s magnetic field is approximately \( 0.5 \times 10^{-4} \ T \), which is \( 5 \times 10^{-5} \ T \). Compare this with \( 2.91 \times 10^{-5} \ T \) calculated from the line current.

Step 5: Determine if Current Affects Compass

Since \( 2.91 \times 10^{-5} \ T \) is less than \( 5 \times 10^{-5} \ T \), the magnetic field from the line is weaker than the Earth's magnetic field. Thus, the effect on the compass is likely not significant.

Key Concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental principle in electromagnetism that allows us to calculate the magnetic field generated by a current-carrying conductor. Specifically, it helps in determining the magnetic field at a certain point in space due to a small segment of current. Using a mathematical expression, this law is written as: \[ B = \frac{\mu_0 I}{2 \pi r} \] where:

• \(B\) is the magnetic field
• \(\mu_0\) is the permeability of free space, approximately \(4\pi \times 10^{-7} \, T \cdot m/A\)
• \(I\) is the current
• \(r\) is the distance from the wire

This law is particularly useful for calculating the magnetic fields of different shapes and sizes of conductors. By utilizing this formula, students can understand how current affects the surrounding magnetic field. This understanding is crucial when analyzing problems involving the interaction of magnetic fields and currents, such as the effect of overhead power lines on compass readings.

Earth's magnetic field

The Earth's magnetic field is a natural magnetic field surrounding the Earth. It is often described as being similar to a giant magnet located at the center of the planet. This field plays an essential role in navigation by providing a reference direction for compasses. The strength of the Earth's magnetic field at the surface typically ranges from about 25 to 65 microteslas (\(\mu T\)), which is equivalent to \(2.5 \times 10^{-5} \, T\) to \(6.5 \times 10^{-5} \, T\). The Earth’s field is vital for many aspects of daily life:

• It protects the planet from solar wind and cosmic radiation.
• It helps migratory animals navigate across long distances.
• It provides a stable direction for compasses used by humans.

In a problem like the one with the transmission line, it's important to compare the artificial magnetic fields created by human activities to the natural magnetic field of the Earth. Thus, understanding the magnitude and influence of these magnetic fields is crucial for ensuring compass accuracy.

Compass accuracy

Compass accuracy can be affected by various factors, including nearby magnetic fields created by electrical currents, such as those in transmission lines. While compasses naturally align themselves with Earth's magnetic field, they can be swayed by stronger nearby magnetic fields. Factors that can affect compass accuracy include:

• Presence of metallic objects that can disrupt the magnetic field lines.
• Proximity to electrical appliances or power lines that generate their own magnetic fields.
• Variations and disturbances in Earth's magnetic field, such as geomagnetic storms.

In our exercise, the hiker suggests walking a small distance to avoid interference from the power line. This suggestion is based on the fact that moving away from the source of a secondary magnetic field can reduce its impact on the compass. Rocking the distances ensures that the dominant field influencing the compass remains the Earth's magnetic field, thus maintaining its accuracy for navigation purposes. Hence, understanding how different magnetic influences affect compasses is essential for accurate navigation.