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?
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 |
| |
| |
| |
| |
| |
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+---------------------||-------------------+
|
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 |
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| |
| |
| |
| |
| | |
+---------------------||-------------------+
|
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Posted by Charlie
on 2016-04-15 19:29:21 |
solution
