Resistance of a Wire Cube: Diagnol Faces

[hamzanaveed]

A wire is broken down into 12 pieces so that each piece is of resistance 1R. The pieces are joined together to form a cube. What would be the resistance at the diagnols of the 4 faces of the cube?

 

[tiny-tim]

welcome to pf!

hi hamzanaveed! welcome to pf!

What would be the resistance at the diagnols of the 4 faces of the cube?

i don't understand what you mean (and also a cube has 6 faces)

anyway, show us what you've tried, and where you're stuck, and then we'll know how to help! ​

 

[yoshtov]

I am assuming you mean the resistance at two diagonal vertices of the cube. Try plotting out all of the different paths that current can take from point A to point B, and then go from there. see if you can put it in the form of a circuit diagram. Drawing a picture of a cube (okay, drawing a 2-dimensional representation of a cube) helps a lot.

 

[A.T.]

Here is a similar one:

A circuit has N nodes. Each node is connected to every other node via a resistor R. What is the total resistance between two nodes?

Spoiler

The trick in both questions is to find the nodes that are equipotential.

 

[PJC01]

The resistance at the diagonals of the four faces of the cube can be calculated by first understanding the overall resistance of the cube. Since each piece of wire has a resistance of 1R, and there are 12 pieces in total, the total resistance of the cube would be 12R.

Now, when the pieces are joined together to form a cube, the resistance is distributed evenly throughout the cube. This means that each face of the cube would have a resistance of 3R (12R divided by 4 faces).

When considering the diagonals of each face, the resistance would follow the same principle. Since the diagonals are formed by pieces of wire that are connected in series, the total resistance of each diagonal would be the sum of the individual resistances. In this case, since there are 3 pieces of wire in each diagonal, the resistance of each diagonal would be 3R (1R+1R+1R=3R).

Therefore, the resistance at the diagonals of the four faces of the cube would be 3R. This can also be confirmed by using the formula for calculating the resistance of a parallel circuit, where the total resistance is equal to the reciprocal of the sum of the reciprocals of each individual resistance. In this case, the total resistance would be 1/ (1/1R+1/1R+1/1R) = 3R.

In summary, the resistance at the diagonals of the four faces of the cube would be 3R, with the overall resistance of the cube being 12R. This demonstrates the importance of understanding the principles of series and parallel circuits when analyzing complex electrical systems.