Resistance between two junctions in a 2D mesh

Click For Summary
SUMMARY

The equivalent resistance between two consecutive junctions in an infinite 2D mesh of conducting wire, where the resistance between any two consecutive junctions is R, is determined to be R/2. This conclusion is reached by applying the principle of superposition of current distribution. When a positive potential E is applied at point A, the current divides equally among the four neighboring junctions, resulting in a total current of i/2 flowing between junctions A and B. The infinite paths available in the mesh do not lead to infinite resistance; rather, they reduce the equivalent resistance.

PREREQUISITES
  • Understanding of electrical resistance and Ohm's Law
  • Familiarity with the principle of superposition in electrical circuits
  • Basic knowledge of infinite series and their convergence
  • Concept of current distribution in electrical networks
NEXT STEPS
  • Study the principle of superposition in more detail, particularly in electrical circuits
  • Explore the concept of equivalent resistance in complex networks
  • Learn about infinite series and their applications in circuit analysis
  • Investigate current distribution in 2D and 3D mesh networks
USEFUL FOR

Electrical engineers, physics students, and anyone interested in circuit analysis and the behavior of electrical networks.

AdityaDev
Messages
527
Reaction score
33

Homework Statement



There is an infinite 2D mesh of conducting wire. Resistance between any two consecutive junctions is R. What is the equivalent resistance between any two consecutive junctions?

Homework Equations

The Attempt at a Solution


Every junction is connected to four points and one of it is connected to the required point. For example, if I start from 0,0 then I have to find the resistance between 0,0 and 1,1. Two wires can go to 1,1 by first moving up and then right or moving right and then up, each step having resistance R. But there are infinite ways to reach 1,1. So wouldn't the resistance be infinite?
 
Physics news on Phys.org
AdityaDev said:

Homework Statement



There is an infinite 2D mesh of conducting wire. Resistance between any two consecutive junctions is R. What is the equivalent resistance between any two consecutive junctions?

Homework Equations

The Attempt at a Solution


Every junction is connected to four points and one of it is connected to the required point. For example, if I start from 0,0 then I have to find the resistance between 0,0 and 1,1. Two wires can go to 1,1 by first moving up and then right or moving right and then up, each step having resistance R. But there are infinite ways to reach 1,1. So wouldn't the resistance be infinite?
The more paths that are available, the lower the equivalent resistance.

The description you give is not clear (to me).

Can you show some image? Also post the complete question as it was given to you.
 
I got to know that you have to use principle of superposition of current distribution to solve the problem. If I Earth the boundary of mesh(at infinity) and apply a positive potential E at A, then I/4 current flows from A to all four neighbouring points and keeps dividing as it moves from A to infinity. So i/4 flows from A to B and potential at B is V.
If I connect a negitive E at B, currenys from infinity flows towards B and finally i/4 flows from A to B. Let potrntial at A be V.
So total current through AB is i/2.
 

Attachments

  • 20150426_085627-1.jpg
    20150426_085627-1.jpg
    25.2 KB · Views: 486

Similar threads

Current in a circuit without external resistance
  • · Replies 8 ·
Replies
8
Views
2K
Two concentric conducting spherical shells and resistor in between
  • · Replies 44 ·
2
Replies
44
Views
5K
Current through an inductor as a function of time
  • · Replies 10 ·
Replies
10
Views
1K
Identical Batteries in Parallel do not Draw Same Current, Non-Ideal?
  • · Replies 7 ·
Replies
7
Views
1K
Calculation of equivalent resistance
  • · Replies 5 ·
Replies
5
Views
2K
Differences in the Brightness of bulbs for 2 separate circuits
  • · Replies 1 ·
Replies
1
Views
3K
Resistance, current and current density
  • · Replies 8 ·
Replies
8
Views
2K
Unraveling the Mystery of EMF Internal Resistance
  • · Replies 2 ·
Replies
2
Views
2K
Finding the Resistance of a Fault in a Pair of Transmission Lines
  • · Replies 8 ·
Replies
8
Views
2K
Underground cable with electrical short
  • · Replies 1 ·
Replies
1
Views
2K