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Hello again. This is Roger Coke Barr for the
Bioelectricity course. We're in Week two, Segment eleven.
And this segment is simply a problem session, involving membrane resistance and
membrane capacitance. I thought I would start off the problem
and then, maybe you could finish it up. Suppose, the question that I ask is, we
have a cylindrical fiber, ten microns in diameter, that has segments that are 100
microns long, you know, micron as in micrometer ten^-6 meters.
It's often shorthand written as a, a [inaudible] followed by an m.
And when people write it, with on the screen, sometimes they write it as and
that's what's happened here in this slide. So, we have a fiber that has, that a
segment of the fiber has those dimensions. We know, for this particular fiber, the cm
is one microfarad per centimeter square, or MS 1500 ohms per centimeter square.
So, for one segment, what is the resistance and the capacitance.
So, the thing you have to know here, just because, you know how to do problems like
this is you have to know that you, you don't attack cm or rm or r or c directly.
You first find the surface area because the surface area comes into both
calculations. So, we would say here that the surface
area for this simple, cylindrical structure, we have this arrangement, this
is 100 micrometers. This diameter is given as ten micrometers
so this will be called that D and I'll call this length L, just to be able to
represent them symbolically. So, I'll have the surface area will be the
circumference. So, that will be pi times the diameter.
And then, to get the area, we'll multiply it times the length.
So, surface area's pi times the diameter, times the length, and if one substitutes
in those numbers, then one gets a numerical answer.
And you have to be careful because our other values are in per centimeter square
to convert A as to centimeter square. Probably, the easiest way to do the
conversion is by putting in value, a value of d and l in centimeter square rather
than putting them in those micrometers. So, if you figure out the relation between
those, I think it's ten^-4. If you put in the relation between those,
then you should get the surface area in centimeter square.
I'll leave it to you to complete that calculation if you would and now, we need
to find the actual resistance and the capacitance for this segment.
As it was explained in earlier, in earlier segments, the calculation itself is quite
easy, resistance will be the membrane resistance.
1500 divided by whatever you found for As, capacitance will be won and result will be
in ohms. Capacitance will be one, microfarad per
centimeter square times As centimeter square.
So, that will be, whatever you find, and that will be in microfarads.
So, as calculations like this are normally done, you could put these numbers together
and you will have the results. I'll ask you to complete these
calculations so that you have the actual number values.
One needs number values like this in doing calculations involving membranes which
always require that you have the actual resistance and the actual capacitance for
the dimensions that are present in a particular problem.
I'd like to just mention a few notes even though these are simple, to come up in
this problem. So, in our notation, what has happened
here is that in some places, such as right here, is, we have cm2, but what's meant by
that is centimeter square. In other words, cm2 means sum of majors
squared. Similar thing happens with Cm, which
really means C subscript m to capacitance of a segment of membrane.
In the olden days, meaning, before computers, people generally got the right
thought, and set these as subscripts and superscripts.
Nowadays, with the advent of computers and computer programs, they often write them
in text, at, on the line, so that, that means that you have to do a mental
translation back and forth between what's written on the line, superscripts, and
subscripts, so as to have everything in this proper notation, that's a second
note. It's helpful to bring to mind the fact
that the units of membrane resistivity are ohm centimeters square.
They're not the same as those for the bulk resistivity.
That is to say the resistivity of the material inside the cell or the material
outside the cell. We used bulk resistivity in Week One and
there, and the units we used there were ohm centimeters.
So, the units are different and then the process one goes through to do the
conversion is different. But then again, the structure is
different. Bulk resistivity exists in three
dimensions where as a membrane is from the viewpoint of resistance, essentially a
flat plane structure, a two-dimensional structure.
And then finally, it's just helpful once again to bring to mind, that as one moves
to get a resistance or a capacitance, one divides, divides Rm by the area to get R
and multiplies Cm by the area to get c. Oh, I wish I had a dollar for every time I
had made this mistake, either on a calculation on paper or on the board or on
a, in a computer program. You just don't think about it.
So, I'm, I'll talk about it here just to give it a little extra impetus.
You'll make the mistake of doing it wrong and dividing both or multiplying both, you
just have to be aware that it's they are different.
And when you come back through to try to find the error, this is one of the places
that the error may exist. So, with this picture of the Duke
University Medical Center, we'll conclude this segment.
We'll comeback in a short while and discuss in the next segment, some of the
conclusions from the second week of work.
Dr. Roger BarrAnderson-Rupp Professor of Biomedical Engineering and Associate Professor of Pediatrics
