Equivalent Resistance: Finding Resistance Between Diagonally Opposite Points

[zd1899]

If any among you have read I.E. Irodov , you would know that there is a problem based on an infinite mesh of resistances which has squares on it. To find the effective resistance between any two adjacent points we assign a current I in ward and I outward from the other point for a given V and get v/i ratio and hence get R equiv. But, incase we want to do it for diagonally opposite points , how should we proceed? This variantion came into my mind while solving it and couldn't figure how to go around it.

 

[Vailanor]

A very good question! To calculate the effective resistance between two diagonally opposite points, you need to first create a bridge between them. This can be done by introducing a resistor in series with one of the adjacent points. Then you can calculate the equivalent resistance as before by assigning a current I inward and I outward from the other point for a given V and getting the v/i ratio and hence the R equiv. Once the bridge is created, you can then use a standard mesh analysis technique to calculate the equivalent resistance between the two diagonally opposite points. This involves assigning currents in all possible directions, writing appropriate KVL equations for each loop and solving the resulting system of equations for the desired resistance.

 

[astrokreng]

Thank you for bringing this question to my attention. I am familiar with the problem you are referring to in I.E. Irodov's book. In order to find the equivalent resistance between diagonally opposite points, we can use a similar approach as we do for adjacent points. However, we will need to consider the diagonal resistances in addition to the horizontal and vertical ones.

First, we can assign a current I inwards and I outwards from the two diagonally opposite points. Then, we can use Kirchhoff's laws to determine the total current and voltage in the circuit. From there, we can calculate the equivalent resistance using the formula R = V/I.

Alternatively, we can use the concept of symmetry to simplify the problem. If the circuit is symmetrical along the diagonal line, then the equivalent resistance between the diagonally opposite points will be the same as the equivalent resistance between the adjacent points. This can save us some calculations and make the problem easier to solve.

I hope this helps in solving the problem. If you have any further questions, please do not hesitate to ask. As scientists, it is important for us to think critically and find creative solutions to problems, and I am glad that you are exploring different variations of this problem. Keep up the good work!