Problem 46 - University Physics with Modern Physics

Question

A nerve signal is transmitted through a neuron when an excess of \(Na^+\) ions suddenly enters the axon, a long cylindrical part of the neuron. Axons are approximately 10.0 \(\mu\)m in diameter, and measurements show that about 5.6 \(\times \space 10^{11} \space Na^+\)ions per meter (each of charge \(+e\)) enter during this process. Although the axon is a long cylinder, the charge does not all enter everywhere at the same time. A plausible model would be a series of point charges moving along the axon. Consider a 0.10-mm length of the axon and model it as a point charge. (a) If the charge that enters each meter of the axon gets distributed uniformly along it, how many coulombs of charge enter a 0.10-mm length of the axon? (b) What electric field (magnitude and direction) does the sudden influx of charge produce at the surface of the body if the axon is 5.00 cm below the skin? (c) Certain sharks can respond to electric fields as weak as 1.0 \(\mu N/C\). How far from this segment of axon could a shark be and still detect its electric field?

Step-by-Step Solution

Answer

(a) 8.96 pC; (b) 0.322 N/C towards axon; (c) 8.97 cm away.

Step 1: Determine Charge Density per Meter

Given that there are \(5.6 \times 10^{11} \ Na^+\) ions per meter and each ion has a charge of \(+e = 1.6 \times 10^{-19} \ C\), the charge per meter is calculated as:\[\lambda = (5.6 \times 10^{11} \ ions/m) \times (1.6 \times 10^{-19} \ C/ion) = 8.96 \times 10^{-8} \ C/m\]

Step 2: Calculate Charge in a 0.10-mm Segment

A 0.10-mm length is equivalent to \(0.10 \times 10^{-3} \ m = 1.0 \times 10^{-4} \ m\). Use the charge density to find the charge in this segment:\[Q = \lambda \times \text{length} = (8.96 \times 10^{-8} \ C/m) \times (1.0 \times 10^{-4} \ m) = 8.96 \times 10^{-12} \ C\]

Step 3: Calculate Electric Field at the Surface of the Body

The electric field due to this point charge at a distance \(r = 5.00 \times 10^{-2} \ m\) is given by Coulomb's law:\[E = \frac{k \cdot Q}{r^2}\]where \(k = 8.99 \times 10^9 \ N \cdot m^2/C^2\). Substitute the values:\[E = \frac{(8.99 \times 10^9) \cdot (8.96 \times 10^{-12})}{(5.00 \times 10^{-2})^2} \approx 3.22 \times 10^{-1} \ N/C\]

Step 4: Determine Maximum Detection Distance for Sharks

Sharks can detect electric fields as weak as \(1.0 \times 10^{-6} \ N/C\). Set this as the electric field and solve for the distance \(r\):\[1.0 \times 10^{-6} = \frac{(8.99 \times 10^9) \cdot (8.96 \times 10^{-12})}{r^2}\]Rearrange and solve for \(r\):\[r^2 = \frac{(8.99 \times 10^9) \cdot (8.96 \times 10^{-12})}{1.0 \times 10^{-6}}\]\[r = \sqrt{80.54 \times 10^{-3}} \approx 8.97 \times 10^{-2} \ m\]Thus, the shark can detect fields up to about 0.0897 meters, or 8.97 cm, away.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle that quantifies the electric force between two charged objects. The law states that the force between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. Mathematically, it can be expressed as:
\[ F = k \frac{{q_1 q_2}}{{r^2}} \]
where \( F \) is the force between the charges, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, \( r \) is the distance between the charges, and \( k \) is Coulomb's constant, approximately equal to \( 8.99 \times 10^9 \, N \cdot m^2/C^2 \).
The law serves as a cornerstone for understanding how electric fields interact, making it crucial when analyzing scenarios such as the electric field produced by charged ions entering a neuron during signal transmission.

Charge Density

Charge density is a measure of the distribution of electric charge per unit length, area, or volume. In this exercise, we focus on linear charge density, represented as \( \lambda \), which describes how charge is distributed along a line, such as an axon.
The linear charge density can be calculated using the formula:
\[ \lambda = \frac{Q}{L} \]
where \( Q \) is the total charge and \( L \) is the length over which the charge is distributed. For example, if there are \( 5.6 \times 10^{11} \) ions per meter, each carrying a charge of \( +e = 1.6 \times 10^{-19} \ C \), the linear charge density becomes essential for finding the charge in any specific segment of the axon.
By understanding charge density, one can better visualize how electric fields are influenced by the continuous flow of charges, especially in biological systems like neurons.

Neuron Electrophysiology

Neuron electrophysiology explores how neurons use electric signals to perform their functions. Neurons, being long and thin cells, transmit information using changes in electric potential across their membranes.
The influx of \( Na^+ \) ions is a crucial component in this process. When these ions enter an axon rapidly, they cause localized changes in the electric potential, representing the nerve signal being transmitted. This change in charge is essentially the movement of electrical energy along the neuron. Understanding how charges move within neurons is central to deciphering how brains process information.
By modeling portions of the axon as point charges, electrophysiology connects micro-level electrochemical events to larger phenomena like nerve impulses, illustrating the complex interaction between biology and electric fields.

Detection of Electric Fields by Sharks

Sharks possess a remarkable sensory ability known as electroreception, which allows them to detect electric fields in their environment. This trait is facilitated by specialized organs called the ampullae of Lorenzini, which can sense minute electric field changes produced by other organisms.
This sensitivity plays a critical role in their hunting behavior, enabling sharks to locate prey even when it is hidden or camouflaged. In relation to axon discharge, sharks could theoretically detect the weak electric fields generated when ions enter a neuron, if they are close enough.
Research has shown that sharks can perceive electric fields as weak as \( 1.0 \, \mu N/C \). The theoretical exercise shows that a shark could detect an axon's electric field from about 8.97 cm away. This insight into how sharks and other species use electric fields highlights the fascinating intersection between biology and physics.