Cylindrical Capacitor Computation

[somasimple]

Hi All,
from this page (Cylindrical Capacitor) =>

http://www.phys.uri.edu/~gerhard/PHY204/tsl105.pdf

What happens when L is <= b? Does the computation change?

and from this page
http://www.physics.umd.edu/perg/abp/TPProbs/Problems/E/E52.htm

The capacitor is simplified to a plane one.
Is it normal that the first equation is ignored, since it doesn't give the same results?

 

[Bob S]

The first equation for a cylindrical capacitor is correct, and is written for an air-filled capacitor (no dielectric).

the equation for a dielectric-filled cylindrical capacitor is

C = 2 pi e_k e₀ L / Ln(b/a) Farads

where e_k == dielectric constant of membrane (unitless)
e₀ == 8.85 x 10^-12 Farads per meter
L == length of capacitor in meters
Ln(x) == natural log of x
b, a == outer and inner radii in meters
=======Homework=====
Check consistency of all units in above equation
Look up the dielectric constant of mylar (DuPont)
Compare to dielectric constant of a myelinated nerve
Look up the description of multiple sclerosis
Look up the definition and description of demyelination

 

[somasimple]

=======Homework=====
Check consistency of all units in above equation
Look up the dielectric constant of mylar (DuPont)
Compare to dielectric constant of a myelinated nerve
Look up the description of multiple myeloma
Look up the definition and description of demyelination

The two last homework lines aren't necessary at all since I'm a health professional.
I understand the first one and may compare the two dielectric constants but myelinated/unmyelinated dielectric vary with authors.

What happens when b grows toward the value of L?

Edit: I found a dielectric value for myelin around 6 to 10.

 

[Bob S]

What happens when b grows toward the value of L?

Edit: I found a dielectric value for myelin around 6 to 10.

RE What happens when b grows toward L? This is a difficult 3-D problem, so I looked in my favorite E&M book (Smythe) but could not find a good answer. My suggestion is if L< b then replace L with b.

I will make a few comments about signal transmission lines and capacitance, and I will try to make them applicable to nerve signals. There are two basic types of analog transmission lines (delay lines) used by engineers: The L - C transmission line (usually dispersionless) and the R - C delay line (usually dispersive). In both types, the signals travel through the series components; L (inductance), or R (resistance), or axons or schwann cells. In all types the capacitance C is between the center conductor (axon or schwann cells) and the outer ground conductor, separated by a space between a and b (insulating dielectric or myalin sheath).

The signal velocity along the center conductor (along axons) is proportional to 1/sqrt(LC) or 1/(RC), and the signal amplitude (voltage or spikes) is proportional to sqrt(L/C) or (1/C). For high amplitude signals and for faster signals, a low capacitance is better. The capacitance for a given geometry (given a, b, L) is proportional to the dielectric constant: Air = 1. mylar (duPont sheets) = 3.2, myelin sheath around axon = 6 to 10, and water = 80. So to preserve both fast nerve signals and high amplitude signals (for given a, b, and L), air is best, followed by mylar, myelin sheath, and water (worst). For a given L and a, larger b is better (but scales only as Ln (a/b))
I hope this helps.

 

[dlgoff]

What happens when L is <= b? Does the computation change?

The electric field was calculated by applying Gauss' law to an infinite cylinder. Which is why L must be >> than b.
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capcyl.html"

 

[somasimple]

The signal velocity along the center conductor (along axons) is proportional to 1/sqrt(LC) or 1/(RC), and the signal amplitude (voltage or spikes) is proportional to sqrt(L/C) or (1/C). For high amplitude signals and for faster signals, a low capacitance is better. The capacitance for a given geometry (given a, b, L) is proportional to the dielectric constant: Air = 1. mylar (duPont sheets) = 3.2, myelin sheath around axon = 6 to 10, and water = 80. So to preserve both fast nerve signals and high amplitude signals (for given a, b, and L), air is best, followed by mylar, myelin sheath, and water (worst). For a given L and a, larger b is better (but scales only as Ln (a/b))
I hope this helps.

Thanks Bob but I simulated the lines with Micro Cap 9 (http://www.spectrum-soft.com/index.shtm" ) and I found effectively the same values of a signal dampening. There is a low pass filtering and, of course, a lower capacitance helps to enhance the cut-off frequency of the filter. But enhancing the limit of the filter doesn't give any mean to enhance the velocity of a signal, I think?
Secondly, the cut-off frequencies are around 10/50 Hz and the the mean sinusoidal frequency (first harmonic) of a firing neuron is often > http://en.wikipedia.org/wiki/Action_potential#Taxonomic_distribution_and_evolutionary_advantages" but it is very well transmitted. It's a kind of huge contradiction.

The electric field was calculated by applying Gauss' law to an infinite cylinder. Which is why L must be >> than b.
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capcyl.html"

That's why I asked