Problem 18 - University Physics with Modern Physics

Question

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 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\(\%\) of the earth's field would have little effect on the bacteria. Take the earth's field to be 5.0 \(\times\) 10\(^{-5}\) T, and ignore the effects of the seawater.)

Step-by-Step Solution

Answer

The magnetic field interferes with bacterial migration within 8 meters of the line.

Step 1: Understand the Problem

We're tasked with finding the range of distances from a transmission line carrying 100 A where the magnetic field interferes with the migration of bacteria. The earth's magnetic field is given as \(5.0 \times 10^{-5}\) T, and the magnetic field at which interference occurs is \(5\%\) of this.

Step 2: Calculate the Threshold Magnetic Field

Calculate \(5\%\) of the earth's magnetic field to determine the threshold for interference.\[5\% \times 5.0 \times 10^{-5} = 2.5 \times 10^{-6} \text{ T}.\]This is the magnetic field we need to compare against.

Step 3: Calculate the Magnetic Field from the Transmission Line

Use the formula for the magnetic field \(B\) from a long straight conductor: \(B = \frac{\mu_0 I}{2\pi r}\), where \(\mu_0 = 4\pi \times 10^{-7} \text{ Tm/A}\) is the permeability of free space, \(I = 100 \text{ A}\), and \(r\) is the distance from the wire.

Step 4: Set Up the Inequality

Set the expression from Step 3 less than or equal to the threshold calculated in Step 2 to find the range of \(r\):\[\frac{4\pi \times 10^{-7} \times 100}{2\pi r} \leq 2.5 \times 10^{-6}.\]

Step 5: Solve the Inequality for r

Simplify the inequality to isolate \(r\):\[\frac{4 \times 10^{-7} \times 100}{2r} \leq 2.5 \times 10^{-6}.\]\[\frac{4 \times 10^{-5}}{2r} \leq 2.5 \times 10^{-6}.\]Solving for \(r\), we find:\[2r \geq \frac{4 \times 10^{-5}}{2.5 \times 10^{-6}}.\]\[2r \geq 16,\]so \(r \geq 8 \text{ meters}\).

Step 6: Conclude the Range of Distances

Thus, the magnetic field from the line is strong enough to interfere if the distance \(r\) is less than or equal to 8 meters from the line.

Key Concepts

Magnetotactic Bacteria

Magnetotactic bacteria are fascinating microorganisms that thrive in aquatic environments. These tiny organisms can navigate using the earth's magnetic field. This ability helps them orient themselves in their habitat, usually to find optimal conditions for survival.
- They contain magnetosomes, which allow them to sense magnetic fields.
- Such bacteria often swim towards the earth's geographic poles, mainly guided by their sensitivity to magnetic features.
The natural movement of these bacteria plays a crucial role in various ecological processes, including biomineralization and the cycling of iron, as they accumulate iron in the form of tiny magnetic particles.

Magnetosomes

Magnetosomes are the key to understanding how magnetotactic bacteria interpret magnetic fields. They are membrane-bound organelles that contain magnetic iron mineral crystals.
These structures are arranged in chains, acting like a compass needle within the bacterial cell.
- The crystals inside are usually made of magnetite (Fe₃O₄) or greigite (Fe₃S₄).
- These minerals provide the bacteria with the ability to align themselves efficiently along magnetic field lines.
Understanding magnetosomes helps us explore their potential applications in nanotechnology and medicine.

Transmission Line

Transmission lines play a crucial role in our electrical networks, carrying electricity across vast distances. These lines generate their own magnetic fields due to the current they transport.
- For a transmission line carrying a current of 100 A, it produces a magnetic field in the surrounding area.
- The strength of this magnetic field diminishes with distance from the line, described by the formula \[B = \frac{\mu_0 I}{2\pi r},\]where \(\mu_0\) is the permeability of free space, \(I\) is the current, and \(r\) is the radial distance.
Knowing this can help engineers and researchers mitigate any potential interference from magnetic fields.

Earth's Magnetic Field

The earth's magnetic field acts as a protective shield around the planet, created by the movement of molten iron in its outer core. This natural magnetic field is vital for life on Earth, influencing animal migration and helping with human navigation historically.
- The strength of the earth's magnetic field at the surface is about \(5.0 \times 10^{-5}\) Tesla.
- In the context of magnetotactic bacteria, the earth's magnetic field serves as a guiding force, affecting their orientation and movement.
Even slight disruptions, such as those from external magnetic sources like transmission lines, can impact these bacteria's natural navigation abilities. Understanding this relationship can aid in designing solutions that minimize environmental and technological conflicts.