How to Calculate Resistivity of a Non-Uniform Wire?

[elionix]

Hello All,

I was curious to know if there is a way to calculate resistivity of a wire that did not have a uniform cross section? For example, what if the cable was in a bow-tie geometry? Is there anyway to quantify the electrical resistivity of the constriction (the area of the cable that has the pinch)?

Thanks!
Elionix

 

[tiny-tim]

Hello Elionix!

… is a way to calculate resistivity of a wire that did not have a uniform cross section?

Resistivity is a property of the material … its shape is irrelevant.

If you mean is it possible to calculate the resistivity if you know the resistance, then yes, just call the resistivity ρ, and treat the cable as a lot of resistors in series, each with length dx and cross-section area A(x).

 

[elionix]

Some follow up questions:

1. What if the resistance was known for, say, material A in series with material B, but both resistivities are unknown. Is there a way to extract the resistivity of material A?

2. What if we knew the resistivity of material A, but not of material B? Knowing the physical dimensions of both materials and also the resistance (A+B in series), is it possible to extract the resistivity of material B?

 

[elionix]

Also, resistivity does change with with the shape of the material, for example look in:
A. Naeemi et al., Proc. IEEE Int. Interconnect Technol. Conf., 183–185 (2008).

Especially in sub-10nm scales, there can be width-dependent resistivity.

Thinking out loud to a response for my questions:

R= ρ1 (A1/L1) + ρ2 (A2/L1)

It seems if I have a wedge shaped configuration, I could just integrate to get the area of interest. so: 1. it's not possible
2. Looks like it is possible to find ρ2 knowing ρ1

 

[tiny-tim]

hi elionix!

(just got up :zzz:)

Some follow up questions:1. What if the resistance was known for, say, material A in series with material B, but both resistivities are unknown. Is there a way to extract the resistivity of material A?

2. What if we knew the resistivity of material A, but not of material B? Knowing the physical dimensions of both materials and also the resistance (A+B in series), is it possible to extract the resistivity of material B?

R= ρ1 (A1/L1) + ρ2 (A2/L1)

It seems if I have a wedge shaped configuration, I could just integrate to get the area of interest. so: 1. it's not possible
2. Looks like it is possible to find ρ2 knowing ρ1

that's right

to put it simply …
1. You have 2 unkowns and only one equation, so no you can't solve it.
2. You have 1 unkown and one equation, so yes you can solve it!

Also, resistivity does change with with the shape of the material, for example look in:
A. Naeemi et al., Proc. IEEE Int. Interconnect Technol. Conf., 183–185 (2008).

sorry, i can't comment on that

 

[Khashishi]

The above equation for R is just an approximation for cases where the length is long compared to sqrt(A) and A is constant. In general, you need to solve an electrostatics problem with boundary conditions. Look at this thread.
https://www.physicsforums.com/showthread.php?t=653837

 

[Khashishi]

If it's anything but a very simple shape, it's a very hairy problem and needs to be solved numerically.