A nerve signal is transmitted through a neuron when an excess of Na^+ ions suddenly enters the a

A nerve signal is transmitted through a neuron when an...

A nerve signal is transmitted through a neuron when an excess of $\mathrm{Na}^{+}$ ions suddenly enters the axon, a long cylindrical part of the neuron. Axons are approximately $10.0 \mu \mathrm{m}$ in diameter, and measurements show that about $5.6 \times 10^{11} \mathrm{Na}^{+}$ ions per meter (each of charge $+e$ ) enter during this process. Although the axon is a long cylinder, the charge does not all enter everywhere at the same time. A plausible model would be a series of point charges moving along the axon. Consider a $0.10 \mathrm{~mm}$ length of the axon and model it as a point charge. (a) If the charge that enters each meter of the axon gets distributed uniformly along it, how many coulombs of charge enter a $0.10 \mathrm{~mm}$ length of the axon? (b) What electric field (magnitude and direction) does the sudden influx of charge produce at the surface of the body if the axon is $5.00 \mathrm{~cm}$ below the skin? (c) Certain shark can respond to electric fields as weak as $1.0 \mu \mathrm{N} / \mathrm{C}$. How far from this segment of axon could a shark be and still detect its electric field?

A nerve signal is transmitted through a neuron when an excess of $\mathrm{Na}^{+}$ ions suddenly enters the axon, a long cylindrical part of the neuron. Axons are approximately $10.0 \mu \mathrm{m}$ in diameter, and measurements show that about $5.6 \times 10^{11} \mathrm{Na}^{+}$ ions per meter (each of charge $+e$ ) enter during this process. Although the axon is a long cylinder, the charge does not all enter everywhere at the same time. A plausible model would be a series of point charges moving along the axon. Consider a $0.10 \mathrm{~mm}$ length of the axon and model it as a point charge. (a) If the charge that enters each meter of the axon gets distributed uniformly along it, how many coulombs of charge enter a $0.10 \mathrm{~mm}$ length of the axon?
(b) What electric field (magnitude and direction) does the sudden influx of charge produce at the surface of the body if the axon is $5.00 \mathrm{~cm}$ below the skin? (c) Certain shark can respond to electric fields as weak as $1.0 \mu \mathrm{N} / \mathrm{C}$. How far from this segment of axon could a shark be and still detect its electric field?

Key Concepts

Electric Charge

Electric charge is a fundamental property of matter that quantifies the amount of excess or deficit of electrons. It is measured in coulombs (C) and is crucial in describing the interactions between particles via the electromagnetic force. In physics, calculations involving the accumulation and distribution of charge are essential for understanding electrostatic phenomena.

Charge Distribution

Charge distribution refers to how electric charge is spread over a region of space. In many cases, such as along a conductor or within a neuronal axon, charge may be distributed uniformly or non-uniformly along a length. Understanding the distribution allows for accurate calculation of local electric fields and potentials by integrating the contributions of each charge element.

Electric Field

The electric field is a vector field representing the force per unit charge exerted on a test charge at any given point in space. It is defined in volts per meter (V/m) or newtons per coulomb (N/C) and is calculated using Coulomb’s law for point charges or by integrating contributions from distributed charges. The concept of the electric field is essential for analyzing the interaction of charges and the propagation of electrical signals.

Coulomb's Law

Coulomb's law provides the quantitative description of the electrostatic force between two point charges. It states that the force between charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This law underpins the calculation of electric fields generated by discrete charges and is fundamental in both theoretical and applied electromagnetism.

Point Charge Approximation

The point charge approximation simplifies the analysis of electrostatic phenomena by modeling a spatially distributed charge as if it were concentrated at a single point. This is particularly useful when the dimensions of the charge distribution are small compared to the distances of interest, allowing for easier computation of the electric fields and potential. It is a common and powerful method used to simplify complex charge geometries.

Transcript

0:00 Hello everyone.

00:01 In this problem we're going to be investigating the electric fields of neurons and how far they can be sensed by certain species of sharks.

00:11 So first we're told that we are we that you know scientists have measured that roughly when an electric signal is transmitted through the neurons you have about 5 .6 10 10 .11 potassium or atrium plus ions per meter and entering the nora.

00:32 So this tells you what is the density of this types of ions, and then, sorry, sodium, sodium density in the entering the ion.

00:45 And then in order to find or convert this to a church density, we can just multiply by the charge of the sodium ions, which is just one e, but positive.

00:55 So this were set to be 5 .6 times 10 to 11 times 1 .6 times 10 to e minus 19.

01:00 Or 8 .96 times the minus 8 coulum meters.

01:05 So this is the charge density of the neurons if the charges were distributed uniformly throughout the entire neuron.

01:15 And so the question is, or part one of the question is, how much of this charge or how much charge can be found given that we model the neuron as a long tube in a given length of the tube, which is 0 .1 0mm millimeters or 1 times 10 to minus 4 meters.

01:35 So all we have to do here is just realize that now we have a charge density, which is like a linear charge density, and we have a length.

01:44 So we just have to multiply those together to get the charge out.

01:48 Because the linear chart density, remember, is just charge over length.

01:52 So dimensionally, you know, this would be what you would naively do in any way, even if you didn't know what the charge density meant.

02:00 Because you looked at it and you would see that you have charge over length.

02:04 So you would have to look by a length to get at the charge.

02:08 So that's kind of what i'm trying to show you here with this little approximation of what's going on.

02:14 So we have the neuron.

02:15 It's got a long tube.

02:16 There are charges all the way through it.

02:18 And assuming they're roughly uniformly distributed, we can then approximate this little length of 0 .1 millimeters as a point.

02:27 Charge of q.

02:30 And so q you can find by multiplying the charge density with length and that works out to be 8 .96 times 10 to minus 12 kuloms...

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