Resistance Cube
- Thread starter physis
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Re: Resistance Cube
By Rattus:
R = sqrt(3)*Rho*L^6
if Rho is the resistance per unit volume and L is the length of one side of the cube.
But one thing is nagging me. A perfect cube would have zero area right at the corner. Nothing to attach the wire to. Same with a sphere. So I keep wondering if the correct answer is R = 0.
Steve
By Rattus:
My first instinct was to integrate the area of the cube over the length between the two corners and multily that by the resistivity. With some very rusty calculus, I came up with:
R = sqrt(3)*Rho*L^6
if Rho is the resistance per unit volume and L is the length of one side of the cube.
But one thing is nagging me. A perfect cube would have zero area right at the corner. Nothing to attach the wire to. Same with a sphere. So I keep wondering if the correct answer is R = 0.
Steve
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Re: Resistance Cube
[ June 13, 2005, 05:19 PM: Message edited by: iwire ]
I am gonna have trouble soldering that.Originally posted by physis:
Using calculus gives you one atom of contact.
Ha Ha.
[ June 13, 2005, 05:19 PM: Message edited by: iwire ]
Re: Resistance Cube
In calculus, there is a particular funnel shape (I forgot what it is called). As you go from the open end toward the point, it tapers in a paticular way that it takes it forever to get to a point. So the funnel has an infinite surface area, but it has a finite volume because of the way it tapers.
If you try to paint the inside with a brush, it would take an infinite amount of paint (because of the infinite surface area). But in theory, you can paint it very easily just by pouring the paint in.
Steve
[ June 13, 2005, 05:23 PM: Message edited by: steve66 ]
In calculus, there is a particular funnel shape (I forgot what it is called). As you go from the open end toward the point, it tapers in a paticular way that it takes it forever to get to a point. So the funnel has an infinite surface area, but it has a finite volume because of the way it tapers.
If you try to paint the inside with a brush, it would take an infinite amount of paint (because of the infinite surface area). But in theory, you can paint it very easily just by pouring the paint in.
Steve
[ June 13, 2005, 05:23 PM: Message edited by: steve66 ]
physis
Senior Member
Re: Resistance Cube
I know it can provide real solutions too. It just seems like with calculus you "tell it what to say".
Edit: is the funnel shape real?
[ June 13, 2005, 05:54 PM: Message edited by: physis ]
I'm just starting to understand calculus, but I get the impression sometimes that it's a mathematical gimmick that, instead of representing reality, is sort of a mathematical band-aid that forces the numbers to do something.
I know it can provide real solutions too. It just seems like with calculus you "tell it what to say".
Edit: is the funnel shape real?
[ June 13, 2005, 05:54 PM: Message edited by: physis ]
Re: Resistance Cube
Pretty soon the funnel tapers to a single molecule. So the funnel can't really follow the gradual, infinite taper down to zero area. And the paint comes in "molecule lumps" which is also ignored. So I guess its more of a mind exercise than something real.
Steve
The thing that "calculus" misses is the thickness of the atoms or molecules that you mentioned. That's what made me think of it.
Pretty soon the funnel tapers to a single molecule. So the funnel can't really follow the gradual, infinite taper down to zero area. And the paint comes in "molecule lumps" which is also ignored. So I guess its more of a mind exercise than something real.
Steve
Re: Resistance Cube
Rattus:
So my hunch about the conductance being zero was correct. But you still have one thing left to explain. I'm stealing the following from one of your posts to another thread:
R= rho x 0/0. This is called an indeterminate form. Can you prove it really is infinity??
Steve
[ June 14, 2005, 09:18 AM: Message edited by: steve66 ]
Rattus:
So my hunch about the conductance being zero was correct. But you still have one thing left to explain. I'm stealing the following from one of your posts to another thread:
The corners have area = 0, but only for a length =0. If we plug this into your equation, we get:
R= rho x 0/0. This is called an indeterminate form. Can you prove it really is infinity??
Steve
[ June 14, 2005, 09:18 AM: Message edited by: steve66 ]
Re: Resistance Cube
Steve,
You cannot apply the resistance formula directly for the reason you note. You have to sneak up on it.
Let us arrange the cube such that one corner lies at the origin of the xyz plane, and the opposite corner lies on the positive x axis.
Now let delta(x) lie between 0 and x on the positive x-axis.
Then delta(R) = k*delta(x)/delta(x)^2 = k/delta(x)
Take the limit of this expression as delta(x) goes to zero, and ZOWIE!, we find that delta(R) goes to infinity.
You can also write a differential equation:
dR = k*x^-2*dx,
where x lies within the 3-sided pyramid with its tip at the origin; then,
R = -k*x^-2 evaluated between 0 and x which leads to,
k/0 - k/x = infinity.
QED
Steve,
You cannot apply the resistance formula directly for the reason you note. You have to sneak up on it.
Let us arrange the cube such that one corner lies at the origin of the xyz plane, and the opposite corner lies on the positive x axis.
Now let delta(x) lie between 0 and x on the positive x-axis.
Then delta(R) = k*delta(x)/delta(x)^2 = k/delta(x)
Take the limit of this expression as delta(x) goes to zero, and ZOWIE!, we find that delta(R) goes to infinity.
You can also write a differential equation:
dR = k*x^-2*dx,
where x lies within the 3-sided pyramid with its tip at the origin; then,
R = -k*x^-2 evaluated between 0 and x which leads to,
k/0 - k/x = infinity.
QED
physis
Senior Member
Re: Resistance Cube
This is hardly electrical but I can't help posting some observations being that this thread is headed into calculus. I'll get to resistance too though.
I stumbled across this thing called Gabriel's Horn Steve. I did'nt make this. I'll post a link if anyone wants it.
You ought to change your limit to the dimension of a copper atom and see what the resistance of copper is.
This is hardly electrical but I can't help posting some observations being that this thread is headed into calculus. I'll get to resistance too though.
I stumbled across this thing called Gabriel's Horn Steve. I did'nt make this. I'll post a link if anyone wants it.
You ought to change your limit to the dimension of a copper atom and see what the resistance of copper is. - Status