How Does Symmetry Solve a Cubic Resistor Network?

[vink]

Dear all,

I'm trying to understand the second figure of the following webpage. The webpage simply solve it by 'symmetry', but I could not figure out its reasoning. Could someone explain how does it work? Thanks in advance.

http://www.schoolphysics.co.uk/age1...ml?PHPSESSID=5b0029c25a5894099c6df916f68d95ac

(b) By symmetry

i1 = 5i2
i3 = 14 i2
i = 24 i2

Therefore:

Total resistance (R) = 7/12 r

 

[haruspex]

The link marked 2i2 can be thought of as shared between two equivalent paths, the 'front' and 'back' paths. So it carries i2 on behalf of each, and since its conductivity is split between them it appears to each to have resistance 2R. A current i1 entering from (say) top left on the front path has two routes to the resistor directly below it. It can go down a single (unlabelled) resistance R or along the sequence R, 2R, R. The former has one quarter the resistance so carries four times the current, 4i2. Now you can add up the currents through different cutsets to find 2i1 = 10i2.

 

[ehild]

I do not understand it either. But you can redraw the circuit by using symmetry. The symmetric points are at the same potential so they can be connected with a wire and considering them a single node. In the second problem, (the third cube in the attachment) the opposite nodes on the upper face of the cube are equivalent, (shown in red) and so are the opposite points on the bottom face (green). Points 4,5 make one node, so resistors (1,4) and (1,5) are parallel. Resistors (2,3) and 2,6) are also parallel. You can redraw the network between A and B and find parallel and series connected resistors, so it is easy the find the resultant resistance. See also https://www.physicsforums.com/showthread.php?t=557461

ehild

 

[vink]

ahhh, thanks for the help, I get it now.

By the way, is anyone aware of any discussion about larger cubic resistor network, such as 3x3, 4x4, nxn etc?

Thanks in advance!

 

[Nickie]

Dear reader,

Thank you for your question. The figure shown in the webpage is known as a cubic resistor network, where the resistors are arranged in a cube formation. In order to understand the reasoning behind the equations given, it is important to first understand the concept of symmetry in physics.

Symmetry refers to the balance, equality or similarity in shape, size or arrangement of an object or system. In this case, the cubic resistor network has a symmetrical arrangement of resistors, meaning that the resistors are all equal in size and shape and are arranged in a balanced manner.

Now, let's look at the equations given:

i1 = 5i2
i3 = 14 i2
i = 24 i2

These equations represent the current flowing through each resistor in the network. By using the concept of symmetry, we can see that the current flowing through the resistors on opposite sides of the cube must be equal. This is because the resistors are arranged symmetrically and therefore, they experience the same conditions and have the same resistance.

For example, i1 represents the current flowing through the top left resistor and i2 represents the current flowing through the bottom right resistor. Since these resistors are on opposite sides of the cube, they must have the same current flowing through them, as shown in the equation i1 = 5i2.

Similarly, the equation i3 = 14 i2 represents the current flowing through the top right resistor and the bottom left resistor, which are also on opposite sides of the cube. Again, due to symmetry, these resistors must have the same current flowing through them.

Finally, the equation i = 24 i2 represents the total current flowing through the entire network. By using the previous two equations, we can see that the total current is equal to 24 times the current flowing through any one resistor, which is represented by i2.

Using Ohm's law (V=IR), we can then calculate the total resistance of the network by dividing the voltage (V) by the total current (i). This gives us the equation R=V/i = (7V)/(24i2) = 7/24 r. This is where the value of 7/12 r comes from in the given solution.

In conclusion, the reasoning behind solving the cubic resistor network by 'symmetry' is based on the equal and balanced arrangement of resistors in the network. By recognizing this symmetry,