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Electrical Cube 3 (Posted on 2016-04-15) Difficulty: 3 of 5
A third wire cube with 1 ohm edges is made, just like the cubes in Electrical Cube and Electrical Cube 2.

This time one of the 1 ohm edges is removed entirely and replaced with a 1 volt battery. How much current is in the wire connecting the battery to the cube structure this time?

See The Solution Submitted by Brian Smith    
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Solution solution Comment 1 of 1
           Each segment except for battery leads
                        is 1 ohm
           Lines consisting of dots are imaginary
      and would have no current flowing even if present

             +------------------------------+
            /.\                            /.\
           / . \                          / . \
          /  .  \                        /  .  \
         /   .   \                      /   .   \
  0 V   /    .    \____________________/    .    \   1 V
       |\    .    /                    \    .    /|
       | \   .   /                      \   .   / |
       |  \  .  /                        \  .  /  |
       |   \ . /                          \ . /   |
       |    \./                            \./    |
       |     +------------------------------+     |
       |                                          |
       |                                          |
       |                                          |
       | 0 ohms        battery            0 ohms  |
       |                   leads                  |
       |                                          |
       |                                          |
       |                                          |
       |                                          |
       |                                          |
       |                     |                    |
       +---------------------||-------------------+
                             |

Since the points at either end of either line consisting of dots are actually one point electrically (that is, one point for each of the dotted lines), The V-shaped half diamond at either end acts as a 1/2 ohm resistor. Between these two 1/2-ohm-equivalent resistors is a path of three parallel connections. Two of these (the top and bottom in the diagram) are simple 1-ohm resistors; the middle one consists of two 1/2-ohm V's similar to the ones near the battery terminal leads, connected by a 1-ohm resistor. So the middle of the three paths that are in parallel is equivalent to a 2-ohm resistor, that is, 1/2 + 1 + 1/2.

So the middle part of the whole circuit is 2 ohms and is in parallel with the two 1-ohm resistors above and below it. The conductance is 1 + 1 + 1/2 = 5/2 mho, making the resistance 2/5 ohm.  The whole circuit's resistance is therefore 1/2 + 2/5 + 1/2 = 7/5.  The current flowing through the circuit, all of which goes through the battery leads, is then 5/7 amp.



                      resistance:
                      
      | 1/2  |            2/5               | 1/2 |
      |      |                              |     |
      |      |                              |     |
      |      |                              |     |
                                                     
             +------------------------------+
            /.\                            /.\
           / . \                          / . \
          /  .  \                        /  .  \
         /   .   \                      /   .   \
  0 V   /    .    \____________________/    .    \   1 V
       |\    .    /                    \    .    /|
       | \   .   /                      \   .   / |
       |  \  .  /                        \  .  /  |
       |   \ . /                          \ . /   |
       |    \./                            \./    |
       |     +------------------------------+     |
       |    5/14 V                         9/14 V |
       |                                          |
       |                                          |
       | 0 ohms -----  battery   -------  0 ohms  |
       |                   leads                  |
       |                                          |
       |                                          |
       |                                          |
       |                                          |
       |                                          |
       |                     |                    |
       +---------------------||-------------------+
                             |


  Posted by Charlie on 2016-04-15 19:29:21
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