Problem 5 - University Physics with Modern Physics

Question

\(Neurons\) 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?

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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: \[ \text{Total ions} = 5.6 \times 10^{11} \text{ ions/m} \times 0.015 \text{ m} = 8.4 \times 10^{9} \text{ ions} \]

Step 4: Calculating Total Charge

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

Step 5: Converting to Scientific Notation

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

Key Concepts

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.