# What characteristics of sheets cause the curved paths between a source and a sink?

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#### jasonbe

##### New Member
What characteristics of sheets cause the paths between a source and a sink to be what they are?

#### MrAl

##### Well-Known Member
Most Helpful Member
Hi,

I think you may want to study electrostatics a little bit to see how charges
diverge naturally. That would help you understand how the charges are set up
in materials like that.

A better idea maybe is to set up the grid of resistors like we talked about in the
other thread. Form a grid of resistors all of the same value, then decrease the
value of each resistor by 1/2 and double the number of resistors (this makes the
grid smaller) and you can also half the length between nodes.
If you keep doing this, you'll eventually have a formula that can tell you about
the resistance between points. It might not be easy to do but you can use
a matrix and look for generalizations as the matrix grows.

You might also look up sheet resistance, and the four point probe method of
measuring sheet resistance.

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#### jasonbe

##### New Member
Hi,

I think you may want to study electrostatics a little bit to see how charges
diverge naturally. That would help you understand how the charges are set up
in materials like that.

A better idea maybe is to set up the grid of resistors like we talked about in the
other thread. Form a grid of resistors all of the same value, then decrease the
value of each resistor by 1/2 and double the number of resistors (this makes the
grid smaller) and you can also half the length between nodes.
If you keep doing this, you'll eventually have a formula that can tell you about
the resistance between points. It might not be easy to do but you can use
a matrix and look for generalizations as the matrix grows.

You might also look up sheet resistance, and the four point probe method of
measuring sheet resistance.

In the grid that you described in your last post, it appeared to me that resistors could be described as connecting small sheets. I think that you had meant for the sheets to be set up in rows and columns - the resistors connecting the rows and columns. I guess I really don't understand that model. I'm not sure what the resistors connecting different rows represented.

In the new model that you proposed in this post, I'm not sure how the length between nodes would be changed, what this represents, and what the different values of the resistors represent.

I looked up sheet resistance and the four-point probe method. Unless they are explained to me otherwise, at this time, I do not consider them to be useful. I may be wrong, but as I understand it this should not be a priority.

I understand that if there is a path between a source and a sink on a sheet, on any one path, the current may take an alternate route by going to either side. However, I'm guessing that one reason why the charge density is greater between the source and the sink is because resistance is a function of distance - and the more curved the path outward from the source and the sink the greater the distance. This is just a guess.

I though of experiments that would involve sheets having different molecular geometries, different resistances, different orientations of molecules with respect to a line segment connecting the source and the sink, different distances between the source and the sink, differently directed magnetic fields with respect to a line segment connecting the source and the sink, and different magnetic strengths. Though, I don't have the materials and wouldn't be able to define the curves of best fit. It is probably better for me to learn about other experiments before I conduct my own. I am still hoping that someone at this site may be able to refer me to one.

I think that I need a formula that relates potential difference to electric field. I have found two or three. But the variables of integration are only the area or volume - not the charge density or electric field. In the case of the sheet, I think that these two variables are changing. In fact, I'm pretty sure that they are. Isn't that why there are multProxy-Connection: keep-alive
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ween the source and the sink? Is there such an equation? Can the charge density and electric field also be made into variables of integration in these formulas - in addition to the area and volume, or is there more to it than that? I'd still need to learn about the how the sheet material would effect these variables first.

I could use help if someone would tell me of any experiments that mathematically described the electric field - if and how it varies in different sheet materials, and how this effects the amount of resistance between the source and the sink. Is there a linear relationship of resistance to distance between a source and a sink?

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#### MrAl

##### Well-Known Member
Most Helpful Member
I guess I really don't understand that model. I'm not sure what the resistors connecting different rows represented.

In the new model that you proposed in this post, I'm not sure how the length between nodes would be changed, what this represents, and what the different values of the resistors represent.

Hi again,

Here's the thing...

If you connect four resistors into a square so that they form four nodes,
you have the beginning of a sheet resistance. It is so far a very crude
model, but still a starting point. The resistance of each resistor
represents a hypothetical incremental resistance along one dimension,
and that's a dimension that has length (or width). Thus, the starting
resistance (say 1 ohm) is the starting incremental resistance and the
length (say 1 inch) is the starting length or width. By calling the lower
left node the origin, we now have a method to not only measure
resistance, but also a method to relate that resistance to actual position
of the four nodes. Thus, resistance becomes a function of position.

Next, we would increase the number of those squares (using more resistors
of course) until we had 4 squares of resistors of squares, except that
all branches are still made from one resistor (not two in parallel).
The other thing we do though, because we are working up to an
infinitely accurate model, is to halve the value of each resistor.
Because we halve the value of each resistor though, we also halve
the length of each resistor. This keeps the same distance between the
four corners of the entire square and so the position still remains relative
to the resistance. The only difference then is that we have now increased
the resolution of the grid, making it more like a sheet.

The next step would again be to double the number of resistors,
halve the resistance and length of each one again, which would increase
the resolution two fold again.

If we keep doing this procedure, eventually we get a really good model
even if we dont use an infinite number of resistors. The resistance
between two points would be related to the position of those points,
and a formula can be developed.

This could be interesting to do even if you dont ever use it.

BTW, what is your goal with this?

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#### jasonbe

##### New Member
Hi again,

Here's the thing...

If you connect four resistors into a square so that they form four nodes,
you have the beginning of a sheet resistance. It is so far a very crude
model, but still a starting point. The resistance of each resistor
represents a hypothetical incremental resistance along one dimension,
and that's a dimension that has length (or width). Thus, the starting
resistance (say 1 ohm) is the starting incremental resistance and the
length (say 1 inch) is the starting length or width. By calling the lower
left node the origin, we now have a method to not only measure
resistance, but also a method to relate that resistance to actual position
of the four nodes. Thus, resistance becomes a function of position.

Next, we would increase the number of those squares (using more resistors
of course) until we had 4 squares of resistors of squares, except that
all branches are still made from one resistor (not two in parallel).
The other thing we do though, because we are working up to an
infinitely accurate model, is to halve the value of each resistor.
Because we halve the value of each resistor though, we also halve
the length of each resistor. This keeps the same distance between the
four corners of the entire square and so the position still remains relative
to the resistance. The only difference then is that we have now increased
the resolution of the grid, making it more like a sheet.

The next step would again be to double the number of resistors,
halve the resistance and length of each one again, which would increase
the resolution two fold again.

If we keep doing this procedure, eventually we get a really good model
even if we dont use an infinite number of resistors. The resistance
between two points would be related to the position of those points,
and a formula can be developed.

This could be interesting to do even if you dont ever use it.

BTW, what is your goal with this?

I might be looking for something more than a crude model. Would you draw a picture? I would like to learn about what causes the curves defining the electric field to be what they are. I've heard from one source that it is not resistivity. I would also like to be able to perform some mathematical operations related to the curves defining the electrical field. I'd like to know of a mathematical description of the electric field, a mathematical description of what the current is along each curve defining the field and how it changes between curves, if the resistance along each curve is directly proportional to the length of the curve or different perhaps because the curve is not straight, what the resistances are along any one curve at different places along the curve - if the resistance is not directly proportional to the length of the curve, if the resistance between the source and sink is directly proportional to the distance between the source and the sink, aProxy-Connection: keep-alive
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what formula relates resistance to the distance between the source and the sink.

#### MrAl

##### Well-Known Member
Most Helpful Member
Hi again,

The reason i went back to the resistor model (which by the way is not a crude
model but the foundation for a very accurate model) is because i felt that that
would fit the bill better than looking at an electrostatic field. It might be good
to look at that anyway, but the thing is that the resistive sheet is a finite object,
whereas the electric field is not confined to a closed space like that. Thus,
we need something that will keep all the currents confined like the resistive
sheet would (assuming you are still interested in resistive sheet characteristics).

Let me see if i can draw this quickly...

Code:
 Four resistors:

o   R   o

R       R

o   R   o

Add more resistors and halve their resistance and the
distance between nodes:

Code:
o R o R o
R   R   R
o R o R o
R   R   R
o R o R o

Each resistor value above is one half of what it was
previously.

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#### jasonbe

##### New Member
Hi again,

The reason i went back to the resistor model (which by the way is not a crude
model but the foundation for a very accurate model) is because i felt that that
would fit the bill better than looking at an electrostatic field. It might be good
to look at that anyway, but the thing is that the resistive sheet is a finite object,
whereas the electric field is not confined to a closed space like that. Thus,
we need something that will keep all the currents confined like the resistive
sheet would (assuming you are still interested in resistive sheet characteristics).

Let me see if i can draw this quickly...

Code:
 Four resistors:

o   R   o

R       R

o   R   o

Add more resistors and halve their resistance and the
distance between nodes:

Code:
o R o R o
R   R   R
o R o R o
R   R   R
o R o R o

Each resistor value above is one half of what it was
previously.

I labeled two of the resistors R1 and R2.
Code:
o  R o R  o
R    R1   R
o  R o R  o
R    R2   R
o  R o R  o

If I were to apply a voltage to the center left and center right sheets - assuming that the "o"s represent sheets, what would R1 and R2 represent? Does the very accurate model represent anything more than this model - or does it just represent things more. Even if models might get very complex, what is known to effect the electric field that they represent?

Could you also refer me to some equations that describe electric fields as a function of specific materials and - even better, properties of materials? Is the relationship between the distance between a source and a sink on a sheet and the amount of resistance between the source and the sink known?

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#### MrAl

##### Well-Known Member
Most Helpful Member
Hi again,

Lets start with this:

o---R---o

That's one resistor between two nodes, not sheets.
This is a very simple resistor.

With this all we know is the voltage applied between the two ends, and
the current that flows. Thus, we know the voltage at a distance d=0
and at a distance d=1, and the current of course, but that's it.

Now we expand:

o-R-o-R-o

Now both R values are made to be 1/2 of what they were before.
Now we have the same for the two ends, and that the distance at
the left end is zero and at the right end is 1 just like before, but now
we have a better model because we now can calculate the center
value of voltage, which is 1/2 of what it is on the right (1/2 in this case).
Thus, we have doubled the resolution of the one dimensional model.

Now we expand again:

o-R-o-R-o-R-o-R-o

with the exception here that this string of resistors would have the same
total distance equal to 1 (it cant be drawn any smaller in pure text).
So lets imagine that we are now zoomed by a factor of 2 also.

What we have now is a better model, because we now know the voltages
of both ends and in the center (as before) but also at a distance of 0.25
and at 0.75 (say inches for simplicity).

If we repeat this process of expanding we end up with a model that can
tell us the voltage at any distance from the left end. The resistance at
that point would be that voltage divided by the current. Or in terms of
distance, with an applied 1v and an initial distance of 1 inch we would have
resistance equals distance divided by the total current.

Each resistor would have a value that we can call the incremental resistance.

#### jasonbe

##### New Member
Hi again,

Lets start with this:

o---R---o

That's one resistor between two nodes, not sheets.
This is a very simple resistor.

With this all we know is the voltage applied between the two ends, and
the current that flows. Thus, we know the voltage at a distance d=0
and at a distance d=1, and the current of course, but that's it.

Now we expand:

o-R-o-R-o

Now both R values are made to be 1/2 of what they were before.
Now we have the same for the two ends, and that the distance at
the left end is zero and at the right end is 1 just like before, but now
we have a better model because we now can calculate the center
value of voltage, which is 1/2 of what it is on the right (1/2 in this case).
Thus, we have doubled the resolution of the one dimensional model.

Now we expand again:

o-R-o-R-o-R-o-R-o

with the exception here that this string of resistors would have the same
total distance equal to 1 (it cant be drawn any smaller in pure text).
So lets imagine that we are now zoomed by a factor of 2 also.

What we have now is a better model, because we now know the voltages
of both ends and in the center (as before) but also at a distance of 0.25
and at 0.75 (say inches for simplicity).

If we repeat this process of expanding we end up with a model that can
tell us the voltage at any distance from the left end. The resistance at
that point would be that voltage divided by the current. Or in terms of
distance, with an applied 1v and an initial distance of 1 inch we would have
resistance equals distance divided by the total current.

Each resistor would have a value that we can call the incremental resistance.

That makes sense. However, in a sheet, there would also be different paths that the current would take. So, there would need to be other paths - as in your other models. However, what would cause the current to take these paths? I'm guessing that the electrons get crowded, repel each other, and then take alternative paths. I'm also guessing - actually I also heard it from one other source, that one reason why current isn't measured to have a more uniform magnitude throughout a sheet is because it is inversely proportional to resistance - which, if I understand correctly, varies according to distance and would therefore increase outward from the point electric contacts. If this is accurate, I don't know how - in a model, these two factors could be represented. Neither do I know how they could be represented in a model in such a way as to result in an amount of current in the model about the same as that in the sheet.

My other post wound up being about different causes of resistance in sheets. I was hoping that this post might be about a specific group of causes associated with this type of sheet - which could include models. I may start another post about the math.

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