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Chapter 17: Problem 6

\(\bullet\) 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?

Short Answer

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

Step by step solution

01

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.
02

Conversion of Length

Since the number of ions is provided per meter, we first convert the length of the axon from centimeters to meters: \[1.5\ \mathrm{cm} = 0.015\ \mathrm{m}\]
03

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:\[5.6 \times 10^{11}\ \text{ions/m} \times 0.015\ \text{m} = 8.4 \times 10^9\ \text{ions}\]
04

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 (\(1.6 \times 10^{-19}\ \text{C}\)): \[Q = 8.4 \times 10^9 \times 1.6 \times 10^{-19} \ \text{C} = 1.344 \times 10^{-9} \ \text{C}\]
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檛 communicate, and the nervous system wouldn鈥檛 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.

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Most popular questions from this chapter

\(\bullet$$\bullet\) Two small plastic spheres are given positive electrical charges. When they are 15.0 \(\mathrm{cm}\) apart, the repulsive force between them has magnitude 0.220 \(\mathrm{N} .\) What is the charge on each sphere (a) if the two charges are equal? (b) if one sphere has four times the charge of the other?

\(\bullet$$\bullet\) Electrophoresis. Electrophoresis is a process used by biologists to separate dif- ferent biological molecules (such as pro- teins) from each other according to their ratio of charge to size. The materials to be separated are in a viscous solution that produces a drag force \(F_{\mathrm{D}}\) propor- tional to the size and speed of the molecule. We can express this relationship as \(F_{11}=K R v,\) where \(R\) is the radius of the molecule (modeled as being spherical), \(v\) is its speed, and \(K\) is a constant that depends on the viscosity of the solution. The solution is placed in an external electric field \(E\) so that the electric force on a particle of charge \(q\) is \(F=q E .\) (a) Show that when the electric field is adjusted so that the two forces (electrical and vis- cous drag) just balance, the ratio of \(q\) to \(R\) is \(K v / E\) . (b) Show that if we leave the electric field on for a time \(T\) , the distance \(x\) that the molecule moves during that time is \(x=(E T / k)(q / R)\) . (c) Sup- pose you have a sample containing three different biological mole- cules for which the molecular ratio \(q / R\) for material 2 is twice that of material 1 and the ratio for material 3 is three times that of mate- rial 1. Show that the distances migrated by these molecules after the same amount of time are \(x_{2}=2 x_{1}\) and \(x_{3}=3 x_{1} .\) In other words, material 2 travels twice as far as material \(1,\) and material 3 travels three times as far as material \(1 .\) Therefore, we have sepa- rated these molecules according to their ratio of charge to size. In practice, this process can be carried out in a special gel or paper, along which the biological molecules migrate. (See Figure 17.60 .) The process can be rather slow, requiring several hours for separa- tions of just a centimeter or so.

\(\bullet\) A positively charged rubber rod is moved close to a neutral copper ball that is resting on a nonconducting sheet of plastic. (a) Sketch the distribution of charges on the ball. (b) With the rod still close to the ball, a metal wire is briefly connected from the ball to the earth and then removed. After the rubber rod is also removed, sketch the distribution of charges (if any) on the copper ball.

\(\bullet\) During a violent electrical storm, a car is struck by a falling high-voltage wire that puts an excess charge of \(-850 \mu C\) on the metal car. (a) How much of this charge is on the inner sur- face of the car? (b) How much is on the outer surface?

\(\bullet$$\bullet\) Point charges of 3.00 \(\mathrm{nC}\) are situated at each of three cor- ners of a square whose side is 0.200 \(\mathrm{m} .\) What are the magni- tude and direction of the resultant force on a point charge of \(-1.00 \mu \mathrm{C}\) if it is placed (a) at the center of the square, (b) at the vacant corner of the square?
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