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Chapter 21: Problem 5

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

Short Answer

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

Step by step solution

01

Understanding the Problem

We need to calculate the total charge entering a 1.5-cm length of the axon when sodium ions (Na+) are entering. The given data is the number of Na+ ions per meter and the charge of each Na+ ion.
02

Identifying the Known Values

We are given: - Charge of one Na+ ion = \(+e\) where \(e = 1.6 \times 10^{-19}\) C - Number of Na+ ions per meter = \(5.6 \times 10^{11}\) ions/m - Length of the axon in meters = 1.5 cm = 0.015 m.
03

Calculating Total Ions Entering the Axon

Calculate the total number of Na+ ions entering a 1.5 cm portion of the axon by multiplying the number of ions per meter by the length in meters: \[ \text{Total ions} = 5.6 \times 10^{11} \text{ ions/m} \times 0.015 \text{ m} = 8.4 \times 10^{9} \text{ ions} \]
04

Calculating Total Charge

Calculate the total charge by multiplying the total number of ions by the charge of one ion: \[ \text{Total charge} = 8.4 \times 10^{9} \text{ ions} \times 1.6 \times 10^{-19} \text{ C/ion} = 1.344 \times 10^{-9} \text{ C} \]
05

Converting to Scientific Notation

The total charge is already in scientific notation, \(1.344 \times 10^{-9}\) C. This is the final answer.

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Key Concepts

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

Neuron
Neurons are specialized cells, also known as nerve cells, that form the fundamental building blocks of the nervous system. They are responsible for receiving sensory input from the external world, processing these signals, and transmitting them to the brain and other parts of the body. This is essential for coordinating actions and responses.
- Neurons communicate through electrical impulses, making them unique in their function.
- Each neuron consists of a cell body (soma), dendrites, and an axon.
- The electrical signals, or impulses, travel along various parts of the neuron, largely focusing on the axon for transmission.
Neurons are crucial for brain function, and their ability to transmit electric impulses quickly and efficiently is key to everything from reflex actions to complex thought. Without neurons, the body wouldn't be able to react to the environment or maintain internal balance.
Axon
The axon is a long, slender projection of a neuron, typically conducting electrical impulses away from the neuron's cell body. It serves as the primary transmission line of the nervous system.
- Axons connect with muscles, glands, and other neurons, facilitating the transfer of messages throughout the body.
- By rapidly transmitting impulses, axons play a vital role in the processing and communication of information in the nervous system.
During impulse propagation, or the movement of an electric signal, sodium ions rapidly enter the axon through specific channels. This shift in ion concentration generates an electric charge, allowing signals to travel rapidly down the length of the axon. The structural composition and length of an axon determine the speed and efficiency at which impulses are transmitted.
Sodium Ions
Sodium ions, denoted as Na\(^+\), are crucial for the generation and propagation of electrical signals in neurons. These ions carry a positive charge, essential for the nerve impulses that enable communication within the nervous system.
- The movement of sodium ions into and out of neurons is a key part of action potential generation.
- When an electric signal arises, sodium channels open, allowing sodium ions to rush into the axon, causing a change in electric charge.
This process is called depolarization and marks the onset of the action potential. The influx of sodium ions is both swift and abundant, illustrating how neurons achieve rapid signaling. Understanding the role of sodium ions highlights their pivotal influence in supporting neuronal communication and responding to stimuli.

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Most popular questions from this chapter

A charge \(q_1 = +\)5.00 nC is placed at the origin of an \(xy\)-coordinate system, and a charge \(q_2 = -\)2.00 nC is placed on the positive \(x\)-axis at \(x = \)4.00 cm. (a) If a third charge \(q_3 = +\)6.00 nC is now placed at the point \(x =\) 4.00 cm, \(y =\) 3.00 cm, find the \(x\)- and \(y\)-components of the total force exerted on this charge by the other two. (b) Find the magnitude and direction of this force.

The earth has a downward-directed electric field near its surface of about 150 N\(/\)C. If a raindrop with a diameter of 0.020 mm is suspended, motionless, in this field, how many excess electrons must it have on its surface?

In a region where there is a uniform electric field that is upward and has magnitude 3.60 \(\times 10^4 \space N/\)C, a small object is projected upward with an initial speed of 1.92 m\(/\)s. The object travels upward a distance of 6.98 cm in 0.200 s. What is the object's charge-to-mass ratio \(q/m\)? Assume \(g = 9.80 \space m/s^2\), and ignore air resistance.

Two thin rods of length \(L\) lie along the \(x\)-axis, one between \(x = \frac{1}{2} a\) and \(x = \frac{1}{2} a + L\) and the other between \(x = -\frac{1}{2} a\) and \(x = -\frac{1}{2} a - L\). Each rod has positive charge \(Q\) distributed uniformly along its length. (a) Calculate the electric field produced by the second rod at points along the positive x-axis. (b) Show that the magnitude of the force that one rod exerts on the other is $$F = {Q^2 \over 4\pi\epsilon_0 L^2} ln [ {(a + L)^2 \over a(a + 2L)} ]$$ (c) Show that if \(a\) \(\gg\) \(L\), the magnitude of this force reduces to \(F = Q^2/4\pi\epsilon_0 a^2\). (\(Hint\): Use the expansion ln \((1 + z) = z - \frac{1}{2} z^2 + \frac{1}{3} z^3 - \cdot\cdot\cdot\), valid for \(\mid z \mid\ll1\). Carry \(all\) expansions to at least order \(L^2/a^2.\)) Interpret this result.

A small sphere with mass \(m\) carries a positive charge \(q\) and is attached to one end of a silk fiber of length \(L\). The other end of the fiber is attached to a large vertical insulating sheet that has a positive surface charge density \(\sigma\). Show that when the sphere is in equilibrium, the fiber makes an angle equal to arctan (\(q\sigma/2mg\epsilon_0\)) with the vertical sheet.
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