Problem 6 - College Physics

Question

\(\bullet\) Signal propagation in neurons. Neurons are components of the nervous system of the body that transmit signals as elec- trical impulses travel along their length. These impulses propa- gate when charge suddenly rushes into and then out of a part of the neutron called an axon. Measurements have shown that, during the inflow part of this cycle, approximately \(5.6 \times 10^{11} \mathrm{Na}^{+}\) (sodium ions) per meter, each with charge \(+e\) enter the axon. How many coulombs of charge enter a 1.5 \(\mathrm{cm}\) length of the axon during this process?

Step-by-Step Solution

Answer

The charge is approximately \(1.344 \times 10^{-9} \ \text{C}\).

Step 1: Understand the Problem

We need to calculate the total charge in coulombs when sodium ions (\(\text{Na}^+\)) enter the axon, given the number of ions per meter and the length of axon in question.

Step 2: Conversion of Length

Since the number of ions is provided per meter, we first convert the length of the axon from centimeters to meters: \[1.5\ \mathrm{cm} = 0.015\ \mathrm{m}\]

Step 3: Calculate Total Number of Ions

Find the total number of sodium ions entering the given length of the axon by multiplying the ions per meter by the length in meters:\[5.6 \times 10^{11}\ \text{ions/m} \times 0.015\ \text{m} = 8.4 \times 10^9\ \text{ions}\]

Step 4: Charge Calculation

Calculate the total charge using the formula for charge, \(Q = n \times e\), where \(n\) is the number of ions and \(e\) is the charge of a single \(\text{Na}^+\) ion (\(1.6 \times 10^{-19}\ \text{C}\)): \[Q = 8.4 \times 10^9 \times 1.6 \times 10^{-19} \ \text{C} = 1.344 \times 10^{-9} \ \text{C}\]

Step 5: Result Interpretation

The total charge entering a 1.5 cm length of the axon is \(1.344 \times 10^{-9} \ \text{C}\). This value indicates the amount of electrical charge transported by the sodium ions into that segment of the axon.

Key Concepts

Axon

The axon is an essential part of a neuron, responsible for transmitting electrical signals over distances within the nervous system. This long, slender projection extends from the neuron's cell body. It's akin to a highway for neural signals, ensuring communication between different parts of the body and brain.
Neurons typically have one axon that can branch at its end. These branches help transmit signals to multiple target cells. The axon's primary role is to carry an electrical impulse away from the neuron's cell body to other neurons, muscles, or glands.
Understanding the structure and function of the axon is crucial in learning about signal propagation, as it is the main avenue through which nerve signals travel.

Electrical Impulses

Electrical impulses, known as action potentials, are rapid signals that travel along the length of the axon. They are the nervous system's way of transmitting information from one place to another. This rapid communication is critical for coordinating actions and responses throughout the body.
An action potential is initiated when a neuron receives a signal, leading to a short-lived reversal of membrane potential. This change is propagated along the axon as an electrical impulse.
Action potentials are essential for everything from simple reflexes to complex thoughts. Without them, neurons couldn’t communicate, and the nervous system wouldn’t function.

Nervous System

The nervous system is a complex network that coordinates the body's voluntary and involuntary actions by transmitting signals to and from different parts of the body. It consists of the central nervous system (brain and spinal cord) and the peripheral nervous system (all other neural elements).
Neurons, including their axons, are integral components of the nervous system. These cells help convey information through electrical impulses. The nervous system detects environmental stimuli, processes this information, and elicits responses.
Its efficiency is vital for everyday tasks, controlling movements, sensing the environment, and even regulating emotions.

Sodium Ions

Sodium ions ( ext{Na}^+ ext{}) play a critical role in the generation and propagation of action potentials in neurons. During an action potential, these ions rapidly enter the neuron, causing a shift in electrical charge that creates an electrical impulse.
This process is facilitated by ion channels that open in response to a signal, allowing sodium ions to flow into the neuron. The inflow of sodium ions changes the electrical charge inside the axon, initiating the action potential.
Once the impulse has traveled, sodium ions are pumped out of the neuron, restoring the original charge and allowing the neuron to ready itself for another signal. This cyclical movement of sodium ions is vital for the nervous system to function seamlessly.