Infinite Grid of Resistors

There is an infinite wire grid with square cells. The resistance of each wire between neighbouring joint connections is R. Find the equivalent resistance of the whole grid between points x and y.

Solution

Normally, resistive network problems are solved when steady-state is reached after the battery is connected. However, for this problem let us find equivalent resistance by transient-state analyses. Resistance is not affected whether it is steady-state or transient-state.

Let us place charge at point x alone. This charge will get distributed equally in four directions. Charge that flows towards y = .

Likewise, let us place charge – at point y alone. This charge will get distributed equally in four directions. Charge that flows towards x = – or charge that flows towards y = +.

When charge is placed at point x and charge – is placed at point y simultaneously, the charge that flows towards y = .


Also,

From (1) & (2), we have .

Now, try to do this problem in similar manner by taking alternate route xaby. You are welcome to share your solution via comment.

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