Equivalent resistance of a cube of resistors

[pc2-brazil]

Homework Statement

Twelve resistors, each one with a resistance of R ohms, form a cube (see attached figure). Find R₁₇, the equivalent resistance of a diagonal of the cube.
(The first attached picture was scanned from the book "Physics" by Halliday, Resnick and Krane, 4th edition.)

Homework Equations

The Attempt at a Solution

I used the fact that, by symmetry, the potential at points 2 and 4 is equal, and the potentials at points 8 and 6 are equal. So, I redrew the resistor configuration and did some simplifications, as the second attached picture shows.
This symmetry argument worked perfectly to show that the resistance between points 1 and 3 (the equivalent resistance of the diagonal of a face) is 3R/4, but it is not working to show that the resistance between points 1 and 7 is 5R/6 (which is the answer given by the book).
I'm stuck at the last simplification attempt shown in the picture.

Thank you in advance.

 

[gneill]

Two suggestions for ways to move forward from your last figure:
1. Apply Y-Δ or Δ-Y transformation.
2. Stick a voltage between points 1 and 7 and find the current it produces (mesh analysis will do nicely).

 

[pc2-brazil]

Two suggestions for ways to move forward from your last figure:
1. Apply Y-Δ or Δ-Y transformation.
2. Stick a voltage between points 1 and 7 and find the current it produces (mesh analysis will do nicely).

I don't know Y-Δ or Δ-Y transformation.
I tried to use mesh analysis. I believe that the current through the circuit would be like the attached figure. But I'm not sure how to proceed.
It seems that it should be possible to use the fact that 2,4 would be at the same potential at 6,8, but I'm not sure why it doesn't work.

 

[gneill]

I don't know Y-Δ or Δ-Y transformation.

It would be worth looking up! Google "Delta-Y Transformation".

I tried to use mesh analysis. I believe that the current through the circuit would be like the attached figure. But I'm not sure how to proceed.

KVL, KCL, the usual stuff.

It seems that it should be possible to use the fact that 2,4 would be at the same potential at 6,8, but I'm not sure why it doesn't work.

It doesn't work because those nodes are NOT at the same potential!

 

[SammyS]

Homework Statement

Twelve resistors, each one with a resistance of R ohms, form a cube (see attached figure). Find R₁₇, the equivalent resistance of a diagonal of the cube.
(The first attached picture was scanned from the book "Physics" by Halliday, Resnick and Krane, 4th edition.)

It looks like you should be finding the resistance across a diagonal of the cube. Which two points are you using as terminals?

 

[ehild]

With respect to points 1 and 7, the three red points are equivalent with each other; and also the three blue points. If a battery is connected between points 1 and 7, points 2,4,and 5 are at equal potential, so can be connected by a wire, and points 3,6,8 are also at the same potential and can be connected.

ehild

 

[pc2-brazil]

It looks like you should be finding the resistance across a diagonal of the cube. Which two points are you using as terminals?

I'm using points 1 and 7 (ignore the points a and b in the first attached picture, because they are not the correct points for the diagonal).

With respect to points 1 and 7, the three red points are equivalent with each other; and also the three blue points. If a battery is connected between points 1 and 7, points 2,4,and 5 are at equal potential, so can be connected by a wire, and points 3,6,8 are also at the same potential and can be connected.

Now I got the correct value 5R/6. I connected points 2, 4 to point 5 and points 6, 8 to point 3 in the first simplification attempt in the attached picture (before, I was only using points 2, 4, 6, 8).

Another question: does the fact that these sets of points are equivalent with each other depend on the chosen points for the resistance (1 and 7)?

Thank you in advance.

 

[ehild]

Another question: does the fact that these sets of points are equivalent with each other depend on the chosen points for the resistance (1 and 7)?

Yes. You need the resistance between points 1 and 7, and it would be different between an other pair of points. Imagine that you determine the resistance by connecting a voltage source of voltage V between the points and measure the current.

The whole arrangement has a three-fold axis of symmetry in your case. The current flowing in at point a or b has three equivalent ways to flow further.
If the source were connected to the neighbouring vertexes like in the original picture, only the pair of red and blue dots would be equivalent.

ehild

 

[Tanya Sharma]

I have the same problem...I am unable to understand how certain points are at same potential using symmetry...

In the resistor cube problem we have to find equivalent resistance between
1) points on the same edge
2) points on face diagonal
3) points on volume diagonal

Please help?

 

[ehild]

If something has symmetry all physical properties have the same symmetry. A cube has a lot of symmetry but connecting a battery to it destroys a lot of them. Same symmetry remains: It is the threefold axis in case when the battery is connected to the terminal points on the body diagonal, and there is a plane of symmetry when the battery is connected to the terminals of an edge or to those of a face diagonal. The equivalent points are of the same colour: They are at the same potential so you can connect them with a short, it does not alter the currents.

ehild

 

[Tanya Sharma]

ehild...Thanks for the explanation. How are we determining what type of symmetry exists by looking at the figure ie line of symmetry or plane of symmetry . Do the equivalent points have to be at equal distance from line of symmetry or plane of symmetry.

Can u explain with another example? I feel i am not getting this symmetry thing...

thanks

 

[ehild]

ehild...Thanks for the explanation. How are we determining what type of symmetry exists by looking at the figure ie line of symmetry or plane of symmetry . Do the equivalent points have to be at equal distance from line of symmetry or plane of symmetry.

Can u explain with another example? I feel i am not getting this symmetry thing...

thanks

Yes, equivalent points have to be at equal distances from the mirror plane or from the rotation axis.
If you perform a symmetry operation on a body, you can not see any difference from the initial body and the one you got.

Look at the picture on the right. It shows how you see the green points 2,4 5 around point 1 seeing from "a". They sit on the vertexes of base of a pyramid. Rotating the whole thing by 120 degrees, only the numbers change, the figure looks the same.

ehild

 

[Tanya Sharma]

Okay...Why haven't we considered a plane of symmetry in the first case instead of a a line of symmetry . The plane of symmetry just like in the second and third case , covering points 1,3,5,7 ?

 

[ehild]

You can, but it will result in a more complicated network. Use the highest symmetry possible.

ehild

 

[Tanya Sharma]

Does that mean first we look for line of symmetry passing through the required terminals across which we have to calculate equivalent resistances... and in case we are not able to find line of symmetry then we should look for plane of symmetry ...just like in the second and third cases we are not able to find line of symmetry.

Does that mean
1) in the first case there is both line of symmetry and plane of symmetry
2) in the second case there is only plane of symmetry
3) in the third case also there is only plane of symmetry ?

 

[ehild]

Does that mean first we look for line of symmetry passing through the required terminals across which we have to calculate equivalent resistances... and in case we are not able to find line of symmetry then we should look for plane of symmetry ...just like in the second and third cases we are not able to find line of symmetry.

Does that mean
1) in the first case there is both line of symmetry and plane of symmetry
2) in the second case there is only plane of symmetry
3) in the third case also there is only plane of symmetry ?

Yes.

And sometimes you get problems with translational symmetry

ehild

 

[Tanya Sharma]

Oh...i just thought i have understood something... but now what is translational symmetry? Please can you give an example so that symmetry concepts are clearer ...thanks

 

[ehild]

Nevermind. It is when you have to get the resistance of an infinite chain. See attachment.

ehild

 

[luiseduardo]

http://www.luiseduardo.com.br/electricity/electrodynamics/cubeofresistors.htm

 

[ehild]

http://www.luiseduardo.com.br/electricity/electrodynamics/cubeofresistors.htm

I saved it, and bookmarked your page. Thanks.

ehild