Length of a Copper Helix

Date: 11/15/98 at 10:35:38
From: Roland Kim
Subject: Calculating the length of two twisted copper wires

Hi Dr. Math,

I have been searching for a formula I could use to determine the length 
of straight wire required if it is to be twisted together with an 
identical piece of wire to yield a helix pair of certain length.

In other words, if I have two pieces of wire that were .040 inch in 
diameter, and they are twisted together so that the distance in one 
complete spiral is 1 inch, and the total linear length of the twisted 
pair is 100 feet, how long does each piece of wire have to be to meet 
this requirement?

I have searched through some textbooks, but I can only find formulas 
that show how to locate points on a helix. If you can point me in a 
particular direction on how to find a solution I would be grateful.  
Thanks.

Roland

Date: 11/16/98 at 12:19:55
From: Doctor Peterson
Subject: Re: Calculating the length of two twisted copper wires

Hi, Roland. If we picture one of your wires making a helix of radius R, 
with one turn taking H inches, it will look like this:

            |<--R-->|
            |       |
      ******|****** |
    **             **
    | ******o****** |----
    |      o|       |   ^
    |     o |       |   |
    |.  o   |       |   |
    ooo.    |       |   |
    |    .  |       |   |
    |      .|       |   |
    |       |.      |   |
    |       |  .    |   |
    |       |    .  |   H
    |       |      .|   |
    |       |       |   |
    |       |       |   |
    |       |       o   |
    |       |       o   |
    |       |      o|   |
    |       |     o |   |
    | ******|****o* |   |
    **      |  o   **   v
      ******o******------

The length of the wire can be found by unwrapping it from the cylinder 
to form a right triangle:

                                                            o
                                                         o  |
                                                      o     |
                                                   o        |
                                                o           |
                                             o              |
                                          o                 |
                                       o                    |
                                    o                       |
                              L  o                          | H
                              o                             |
                           o                                |
                        o                                   |
                     o                                      |
                  o                                         |
               o                                            |
            o                                               |
         o                                                  |
      o                                                     |
    *********************************************************
                              2 pi R

We'll get:

    L = sqrt(H^2 + (2 pi R)^2)

as the length of one turn of wire, so the ratio of the length of the 
twisted pair to the length of the wire will be:

    L / H = sqrt(1 + 4 pi^2 (R/H)^2)

The radius of the cylinder about which the center of the wire will 
spiral will be about the same as the radius of the wire itself:

           ********   oooooooo  ********
        ***        ooo       *ooo       ***
      **         oo   **   **    oo        **
     *          o       * *        o         *
    *          o         *    R     o         *
    *          o         *----------o         *
    *          o         *          o         *
     *          o       * *        o         *
      *          o     *   *      o         *
       ****       oooo*     **oooo      ****
           ********   oooooooo  ********

So in your example, with R = .020 in and H = 1 in, we get:

    L / H = sqrt(1 + 4 pi^2 .020^2) = 1.0079

and for a 100 ft pair, each wire is 1.0079 * 100 ft = 100.79 ft.

That doesn't sound like much difference. It will increase if the wires 
don't touch tightly all the way around. You may have to adjust the 
radius to correspond to reality!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/