Problem 44 - College Physics

Question

Magnetic sensitivity of electric fish. In a problem dealing with electric fish in Chapter 19, we saw that these fish navigate by responding to changes in the current in seawater. This current is due to a potential difference of around 3.0 \(\mathrm{V}\) generated by the fish and is about 12 \(\mathrm{mA}\) within a centimeter or so from the fish. Receptor cells in the fish are sensitive to the current. since the current is at some distance from the fish, the sensitivity of these cells suggests that they might be responding to the magnetic field created by the current. To get some estimate of how sensitive the cells are, we can model the current as that of a long, straight wire with the receptor cells 2.0 \(\mathrm{cm}\) away. What is the strength of the magnetic field at the receptor cells?

Step-by-Step Solution

Answer

The magnetic field strength is approximately \(2.4 \times 10^{-7} \, \text{T}\).

Step 1: Identify Known Values

We know the current, \( I = 12 \, \text{mA} = 0.012 \, \text{A} \) and the distance from the wire, \( r = 2.0 \, \text{cm} = 0.02 \, \text{m} \). We will use these to find the magnetic field strength.

Step 2: Recalling the Magnetic Field Formula

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

Step 3: Calculate the Magnetic Field Strength

Substitute the values for \( I \), \( r \), and \( \mu_0 \) into the formula:\[B = \frac{{\left(4\pi \times 10^{-7} \right) \times 0.012}}{{2\pi \times 0.02}}\]Simplifying, we have:\[B = \frac{{4 \times 10^{-7} \times 0.012}}{{0.02}} = \frac{{4.8 \times 10^{-9}}}{{0.02}} = 2.4 \times 10^{-7} \, \text{T}\]

Step 4: Conclusion

The strength of the magnetic field at the receptor cells located 2 cm away from the fish's current is approximately \(2.4 \times 10^{-7} \, \text{T}\).

Key Concepts

Magnetic field calculation

The process of calculating the magnetic field generated by electric currents is a fundamental concept in electromagnetism. When a current flows through a conductor, such as a wire, it generates a magnetic field around it. This phenomenon can be described using Ampère's Law.
To estimate the magnetic field strength created by a current, the formula used is:

• \( B = \frac{{\mu_0 I}}{{2\pi r}} \)

This formula is specifically for an infinitely long, straight wire. Here, \( B \) is the magnetic field strength, \( \mu_0 \) is the permeability of free space, \( I \) is the current, and \( r \) is the distance from the wire.
The formula indicates that the magnetic field strength is directly proportional to the current and inversely proportional to the distance from the conductor. This means that the closer you are to the source of the current, the stronger the magnetic field, and vice versa. This is crucial in applications such as understanding electric fish navigation, where sensitivity to magnetic fields can influence behavior.

Biophysics

Biophysics integrates physical principles with biological systems to understand how organisms function at a physical level. In the context of electric fish, biophysics helps explain how these creatures detect and utilize electrical and magnetic fields for navigation and communication.
Electric fish possess specialized receptor cells that are sensitive to changes in electric and magnetic fields. These receptors can detect external stimuli and convert them into bio-electrical signals, allowing fish to interpret their environment effectively. Such abilities are vital for navigation, finding prey, and avoiding predators in a murky aquatic environment.
The study of electric fish is a perfect case of biophysics, linking the electrical phenomena produced by living tissues to the physical principles like electromagnetism. This fascinating integration highlights how specific organisms have evolved mechanisms to exploit fundamental physical properties for survival.

Permeability of free space

The permeability of free space, denoted as \( \mu_0 \), is a fundamental constant in electromagnetism. It measures the ability of the vacuum to support the formation of a magnetic field and is essential in calculations involving magnetic fields and forces.
The value of the permeability of free space is precisely defined as \( 4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A} \). This constant appears in the formula for calculating the magnetic field around a current-carrying conductor, highlighting its importance in defining the relationship between current and magnetic fields.
Understanding \( \mu_0 \) is crucial when examining electromagnetic phenomena, as it helps predict how magnetic fields behave in different environments. For instance, in the case of a wire carrying an electric current, \( \mu_0 \) allows us to quantify the magnetic influence of the current over a distance, which is critical for myriad applications in technology and natural phenomenon exploration.