Problem 9 - University Physics with Modern Physics

Question

Transmission of Nerve Impulses. Nerve cells transmit electric signals through their long tubular axons. These signals propagate due to a sudden rush of \(\mathrm{Na}^{+}\) ions, each with charge \(+e,\) into the axon. Measurements have revealed that typically about \(5.6 \times 10^{11} \mathrm{Na}^{+}\) ions enter each meter of the axon during a time of 10 \(\mathrm{ms}\) . What is the current during this inflow of charge in a meter of axon?

Step-by-Step Solution

Answer

The current is \( 8.96 \ \mu\mathrm{A} \).

Step 1: Understand the Given Data

We are given that the number of \( \mathrm{Na}^{+} \) ions entering each meter of the axon is \( 5.6 \times 10^{11} \), and this occurs over a time period of 10 milliseconds (\( 10 \ \mathrm{ms} = 10 \times 10^{-3} \ \mathrm{s} \)). Each ion carries a charge of \( +e \), where \( e \) (the elementary charge) is approximately \( 1.6 \times 10^{-19} \ \mathrm{C} \). We need to calculate the current.

Step 2: Calculate the Total Charge

The total charge \( Q \) entering the axon is determined by multiplying the number of ions by the charge of one ion. Therefore, \( Q = \text{{number of ions}} \times e = 5.6 \times 10^{11} \times 1.6 \times 10^{-19} \ \mathrm{C} \).

Step 3: Calculate the Current

Current \( I \) is defined as the total charge \( Q \) flowing per unit time \( t \). We use the formula \( I = \frac{Q}{t} \). Substitute for \( Q \) and \( t \, (10^{-2} \ \mathrm{s}) \) to find \( I \).

Step 4: Perform the Calculations

First, calculate the total charge: \( Q = 5.6 \times 10^{11} \times 1.6 \times 10^{-19} \ = 8.96 \times 10^{-8} \) C. Then, calculate the current: \( I = \frac{8.96 \times 10^{-8}}{10 \times 10^{-3}} = 8.96 \times 10^{-6} \ \mathrm{A} \), or \( 8.96 \ \mu \mathrm{A} \).

Key Concepts

Electric Current in Nerve Axons

Nerve axons are fascinating structures that transmit electric signals crucial for nervous system communication. The signal transmission relies on the movement of charged particles. In the context of nerve axons, electric current refers to the flow of these charged sodium ions \(\mathrm{Na}^{+}\) through the axon. This occurs when nerve impulses are transmitted.
Understanding current in this scenario involves recognizing that when ions move into the axon, they collectively carry a net charge which constitutes an electric current. Current, denoted by \(I\), is essentially the rate at which this charge flows, given by the formula \(I = \frac{Q}{t}\). Here, \(Q\) represents the total charge, and \(t\) is the time over which the ions flow into the axon.
By examining this process, we can appreciate how biological systems utilize physics to perform complex functions like nerve impulse transmission.

Sodium Ions in Nerve Signals

In nerve signal transmission, \(\mathrm{Na}^{+}\) ions play a critical role. When a nerve impulse occurs, there is a sudden influx of \(\mathrm{Na}^{+}\) ions into the axon. This rapid movement of sodium ions is fundamental to the propagation of electrical signals.
Nerve cells use a mechanism known as the sodium-potassium pump to maintain a high concentration of sodium ions outside the axon relative to the inside. This creates a gradient that, when disturbed, allows sodium ions to rush into the axon through specific channels.

• This influx changes the electrical potential across the cell membrane, initiating an action potential.
• This action potential travels along the axon to communicate signals to other nerve cells or muscles.

The interaction between sodium influx and the action potential is a prime example of electrophysiology, where electrical phenomena within biological systems are studied to understand nervous system function.

Elementary Charge Calculation

Calculating the charge involved in nerve impulse transmission requires understanding elementary charge. The elementary charge, denoted by \(e\), is a fundamental physical constant roughly equal to \(1.6 \times 10^{-19} \, \mathrm{C}\). This charge represents the amount of charge carried by a single proton or electron.
In the context of \(\mathrm{Na}^{+}\) ions, each ion carries a charge equivalent to \(+e\). To find the total charge \(Q\) moving into the axon, multiply the number of ions by the elementary charge:
\[Q = \text{{number of ions}} \times e = 5.6 \times 10^{11} \times 1.6 \times 10^{-19} \, \mathrm{C}\]
Therefore, the charge entering per meter of the axon is \(8.96 \times 10^{-8} \, \mathrm{C}\). This calculation is crucial for determining the current, emphasizing the importance of understanding fundamental constants like the elementary charge in real-world applications.