# Density Variation of a Bent Wire

Thread Starter

#### MrAl

Joined Jun 17, 2014
8,066
Let's say we have a straight copper wire circular cross section say #10 AWG which is about 0.1 inch diameter. We bend it into a complete and perfect circle about 2 inches in mean diameter.
Now the inner diameter is about 1.95 inches, while the outer diameter is about 2.05 inches.
What is the change in density across it's cross section from inner to outer diameter?

Caveats:
1. For one, the circular cross section may turn into a slight oval.
2. Since the cross section density is not uniform we need a formula for the cross section density change, or for a change in cross sectional shape.

This inadvertently came up in another thread even though it was not really part of the origina problem.

#### SamR

Joined Mar 19, 2019
3,197
Based on my machine shop experience cold bending iron, the outside stretches, and the inside crimps. If stretched the metal "fibers" separate and the crimped metal bulges and distorts because the metal density cannot increase or decrease. When heated, the metal "flows" into shape maintaining its density. So the density cannot change as the shape changes. You cannot cold bend a 2" x 4" cross-section of iron 90° without "breaking" but you can do it when heated "red hot". There is a temperature at which any metal retains it's plastic state and is malleable.

#### oz93666

Joined Sep 7, 2010
737
This is quiet easy if we assume the diameter of the wire 0.1in doesn't change (it may do).... the circumference of bend is proportional to the diameter ...

In the bent case the density along the inner curve will be standard density of copper all the rest has been irreversibly stretched and so has a lower density ... so atoms covering a distance of 1.95 , on the inner circumference will cover a length of 2.05 on outer circumference ....

1.95/ 2.05 = 0.9512 ........... So the density at outer circumference will be 0.9512 of normal

Along the axis of the wire the density loss will be half that ... and so on ....

I expect all this is wildly out because the diameter of wire will change , as will shape into an oval

Not clear how this will effect resistance .

#### AlbertHall

Joined Jun 4, 2014
11,129
When wire is being made is it drawn out from an ingot down to whatever size is required. If stretching wire changed its density then wires of different sizes would have different densities depending how much they had been stretched. This doesn't happen so stretching doesn't change the density. I suspect that what happens is that the shape changes to accommodate the different circumference on the inside and outside of the bend.

• shortbus and SamR
Thread Starter

#### MrAl

Joined Jun 17, 2014
8,066
When wire is being made is it drawn out from an ingot down to whatever size is required. If stretching wire changed its density then wires of different sizes would have different densities depending how much they had been stretched. This doesn't happen so stretching doesn't change the density. I suspect that what happens is that the shape changes to accommodate the different circumference on the inside and outside of the bend.
Hi,

The main idea of this thread really is what happens to the inductance of the curved wire.
If the shape changes where there is less material on the inside diameter and more on the outside, then that would mean the inside is closer to the center now and the outer part is farther out from the center. That would mean the inductance would drop ever so slightly.
Yeah it is hard to predict the change in shape.

#### oz93666

Joined Sep 7, 2010
737
Hi,
The main idea of this thread really is what happens to the inductance of the curved wire.
If the shape changes where there is less material on the inside diameter and more on the outside, then that would mean the inside is closer to the center now and the outer part is farther out from the center. That would mean the inductance would drop ever so slightly.
Yeah it is hard to predict the change in shape.
See this question ..."Does electricity travel through the wire or on the outer surface?" ...seems it tends to move on the outer surface of the wire
http://www.engineering.com/Ask@/qactid/1/qaqid/1672.aspx

Thread Starter

#### MrAl

Joined Jun 17, 2014
8,066
See this question ..."Does electricity travel through the wire or on the outer surface?" ...seems it tends to move on the outer surface of the wire
http://www.engineering.com/Ask@/qactid/1/qaqid/1672.aspx
Hi,

Not sure how you came to that conclusion but that cant be right for a number of reasons.
Number 1 is that when we choose a wire diameter we are really choosing a cross sectional area that can handle the current we are working with, and if we double the diameter we triple the current handling ability.
That factor of 3 is because the area (a two dimensional quantity) changes in two directions. A circular ring around the wire would only increase 2 times. Thus if the current only traveled near the surface we'd only get a 2 fold inrease in current handling ability instead of 3.
But that is of course for DC or very low frequency. For higher frequencies it does travel closer to the surface and this is noted as the skin effect.

#### AlbertHall

Joined Jun 4, 2014
11,129
Number 1 is that when we choose a wire diameter we are really choosing a cross sectional area that can handle the current we are working with, and if we double the diameter we triple the current handling ability.
Triple? Quadruple, surely. Double the diameter means quadruple the area and so quadruple the current (at DC or low frequency so no skin effect).

#### shortbus

Joined Sep 30, 2009
8,441
Hi,

The main idea of this thread really is what happens to the inductance of the curved wire.
If the shape changes where there is less material on the inside diameter and more on the outside, then that would mean the inside is closer to the center now and the outer part is farther out from the center. That would mean the inductance would drop ever so slightly.
Yeah it is hard to predict the change in shape.
Isn't that why they use the "mean" diameter of the circle or shape for those calculations? I know from experience when designing a stamping die to make a part from sheet metal that is the length you start with to make a prototype part.

Thread Starter

#### MrAl

Joined Jun 17, 2014
8,066
Triple? Quadruple, surely. Double the diameter means quadruple the area and so quadruple the current (at DC or low frequency so no skin effect).
Hi,

Sorry i meant go up in wire size by 2 which really means go down in wire number by 2.
Yes double the diameter means double the radius and so we see r^2 ratio come out to 2^2=4 yes.
I'll have to fix that post thanks.

Thread Starter

#### MrAl

Joined Jun 17, 2014
8,066
Isn't that why they use the "mean" diameter of the circle or shape for those calculations? I know from experience when designing a stamping die to make a part from sheet metal that is the length you start with to make a prototype part.
Not sure what you mean. The mean diameter is the diameter, but if you mean for the inductance calculation the mean may still depend on shape due to the way the mean is calculated, I'd have to review that.

#### shortbus

Joined Sep 30, 2009
8,441
Not sure what you mean. The mean diameter is the diameter, but if you mean for the inductance calculation the mean may still depend on shape due to the way the mean is calculated, I'd have to review that.
From our own site here - https://www.allaboutcircuits.com/tools/coil-inductance-calculator/
"D" is the mean diameter, in most things other than this I guess. They call it "loop diameter".

This is what is used for metal working designs so assumed it was called the same in an inductor formula - https://sielearning.tafensw.edu.au/toolboxes/toolbox905/3_tem/tem_t6/htm/tem6_2_2.htm

Thread Starter

#### MrAl

Joined Jun 17, 2014
8,066
From our own site here - https://www.allaboutcircuits.com/tools/coil-inductance-calculator/
"D" is the mean diameter, in most things other than this I guess. They call it "loop diameter".

This is what is used for metal working designs so assumed it was called the same in an inductor formula - https://sielearning.tafensw.edu.au/toolboxes/toolbox905/3_tem/tem_t6/htm/tem6_2_2.htm
Hi,

Perhaps but for more complex shapes a summation is used that involves summing every point in the cross section to every other point, so it's very different. I'll see if i can find that info again.

It's called the "geometric mean diameter" but the calculation is not simple. I once used a Monty Carlo method which involves distances between random points and the average sum.

Last edited:
• shortbus