Problem 48 - University Physics with Modern Physics

Question

BI0 Electric Field of Axons. 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\(\mu \mathrm{m}\) in diameter, and measurements show that about \(5.6 \times 10^{11} \mathrm{Na}^{+}\) 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. Let us look at 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 \(\mathrm{cm}\) below the skin? (c) Certain sharks can respond to electric fields as weak as 1.0\(\mu \mathrm{N} / \mathrm{C}\) . How far from this segment of axon could a shark be and still detect its electric field?

Step-by-Step Solution

Answer

(a) Charge is 8.96 x 10^{-12} C. (b) Electric field is 3.22 x 10^3 N/C. (c) Shark detection range is 8.47 m.

Step 1: Calculate the Total Charge Entering a 0.10 mm Segment

First, we need to figure out how many charges enter a 0.10 mm segment of the axon.Given:- Charge density along 1 meter of axon: \( 5.6 \times 10^{11} \text{ Na}^{+} \) ions/m.- Charge of each \( \text{Na}^+ \) ion: \( e = 1.6 \times 10^{-19} \text{ C} \).Calculate the number of ions in 0.10 mm (which is 0.0001 m):\[\text{Number of ions} = 5.6 \times 10^{11} \text{ ions/m} \times 0.0001 \text{ m} = 5.6 \times 10^{7} \text{ ions}.\]Now, calculate the total charge:\[Q = 5.6 \times 10^{7} \text{ ions} \times 1.6 \times 10^{-19} \text{ C/ion} = 8.96 \times 10^{-12} \text{ C}.\]

Step 2: Calculate the Electric Field at the Surface of the Body

The total charge \( Q \) we found earlier acts like a point charge producing an electric field:For a point charge, the electric field \( E \) at a distance \( r \) is:\[E = \frac{kQ}{r^2},\]where \( k \) is Coulomb's constant \( 8.99 \times 10^9 \text{ N m}^2/\text{C}^2 \).Given the axon is 5.0 cm (or 0.05 m) below the skin:\[E = \frac{8.99 \times 10^9 \times 8.96 \times 10^{-12}}{0.05^2} \approx 3.22 \times 10^3 \text{ N/C}.\]

Step 3: Determine Detection Range for a Shark

Certain sharks can detect electric fields \( E_{\text{shark}} \) as weak as \( 1.0 \mu \text{N/C} = 1.0 \times 10^{-6} \text{ N/C} \).The detection distance \( r \) is given by rearranging the point-charge electric field equation:\[E_{\text{shark}} = \frac{kQ}{r^2} \implies r = \sqrt{\frac{kQ}{E_{\text{shark}}}}.\]Plug in values to find \( r \):\[r = \sqrt{\frac{8.99 \times 10^9 \times 8.96 \times 10^{-12}}{1.0 \times 10^{-6}}} \approx 8.47 \text{ m}.\]

Key Concepts

Axon

An axon is a crucial part of a neuron that plays a pivotal role in the transmission of nerve signals. Think of it as a long, slender tube that extends from the neuron, making it possible for signals to travel quickly over long distances.
Its primary function is to carry electrical impulses away from the neuron's cell body. This allows the neuron to communicate with other neurons, muscles, or glands.

• The axon is integral for quick and efficient signal transfer.
• This enables bodily functions to be coordinated and responses to be rapid.

In our context, when sodium ions (Na⁺) enter the axon, they cause the generation of an electric current that facilitates this communication.

Neurons

Neurons are the fundamental units of the brain and nervous system. They are the cells responsible for receiving sensory input from the external world, processing this information, and sending commands to muscles and glands.
Each neuron consists of three main parts: the cell body, dendrites, and the axon. Dendrites are the branch-like structures that receive messages from other neurons, while the axon transmits signals.
In our exercise, neurons demonstrate their ability to generate electric signals by permitting sodium ions to move into the axon.

• This influx creates an electrochemical signal.
• The neuron then transmits this signal along its axon to other neurons or to target cells.

Coulomb's Law

Coulomb's Law is a fundamental principle in physics that describes the force between two charged objects. It states that the electric force between charged objects is directly proportional to the product of the sizes of their charges and inversely proportional to the square of the distance between their centers.
In simpler terms, if you have two charges, the force between them becomes stronger as they get closer and weaker as they move apart.

• The formula is: \[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \] where \( F \) is the force, \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are charges, and \( r \) is the distance.

The step-by-step solution of our problem uses this law to determine the electric field produced by the charges moving along the axon.

Electric Field Detection

Electric field detection is a fascinating capability in certain species, particularly in some aquatic animals like sharks. These animals can sense even very weak electric fields generated by movements of ions such as those across an axon.
Electric fields are invisible forces that arise from electric charges, and their detection allows predators to locate prey or sense dangers in their environment.

• Sharks use special sensors called ampullae of Lorenzini to detect these fields.
• This detection is precise, enabling them to sense fields as weak as 1.0 \( \mu \text{N/C} \).

In the problem context, electric field detection is explored by calculating how far a shark could be from the charged axon segment and still perceive its field.

Nervous System Physiology

The physiology of the nervous system encompasses the complex processes that allow the body to respond to stimuli. It involves the coordination between neurons and their axons, synapses, and neurotransmitters.
This intricate network ensures communication between different parts of the body, integrating sensory information and responding with actions or adjustments.

• Activation of neurons, such as those in our problem, involves electrical impulses.
• The movement of ions like sodium (\( \text{Na}^+ \)) into and out of axons is crucial for this signal transmission.

Understanding these concepts provides insight into how vital responses such as muscle movements or reflexes are initiated and regulated.