Question

A nerve signal is transmitted through a neuron when an excess of Na+ ions suddenly enters the axon, a long cylindrical part of the neuron. Axons are approximately 10 μm (10x10-6 m) in diameter, and measurements show that about 5x1011 Na+ ions per meter (each of charge +e) enter during this process. Although the axon is a long cylinder, the charge do not all enter the axon at the same time. A plausible model would be a series of point charges moving along the axon. Let us look at a 0.10 mm length of the axon as a point charge. a) If the charge that enters each meter of the axon gets distributed uniformly along it, how many Coulomb of charge enter a 0.10 mm length of the axon? b) What is the magnitude of the electric field produced by a 0.10 mm length of the axon at the surface of the body (of a fish for example) if the axon is 5.00 cm below the skin (assume it is small enough to be considered a point charge)? c) Certain sharks can respond to electric fields as weak as 1.0 μN/C (sharks have special detectors called ampullae of Lorenzini for this purpose), how far from the 0.10 mm length of the axon (i.e. from the fish) could a shark be and still detect its electric field (assume you can consider the axon as a point charge)?

Answer 1

given

5 x 10¹¹ Na⁺ ions per meter

so now for 0.1 mm = 0.1 x 10^-3 m length

n = 5 x 10¹¹ x 0.1 x 10^-3

= 5 x 10⁷ Na⁺ ions

then

a)

the charge is Q

Q = 5 x 10⁷ x 1.6 x 10^-19

Q = 8 x 10^-12 C Coulomb of charge enter a 0.10 mm length of the axon.

b )

to find electric field

E = k Q / r²

= 9 x 10⁹ x 8 x 10^-12 / 0.05²

E = 28.8 N/C

the magnitude of the electric field produced by a 0.10 mm length of the axon

at the surface of the body is E = 28.8 N/C

c )

E = 1 uN/C

= 1 x 10^-6 N/C

to find the distance " r "

E = k Q / r²

1 x 10^-6 = 9 x 10⁹ x 8 x 10^-12 / r²

r² = 9 x 10⁹ x 8 x 10^-12 / 1 x 10^-6

= 72000

r = 268.328 m far from the 0.10 mm length of the axon

(i.e. from the fish) could a shark be and still detect its electric field.