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in Physics by (75.0k points)

There is an infinite wire grid with square cells (Fig. 3.38). The resistance of each wire between neighbouring joint connections is equal to R0. Find the resistance R of the whole grid between points A and B. 

Instruction. Make use of principles of symmetry and superposition.

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Suppose that the voltage V is applied between the points A and B then 

V = IR = I0 R0

where R is resistance of whole the grid, l0 the current through the grid and I ,the current through the segment AB. Now from symmetry, I/4 is the part of the current, flowing through all the four wire segments, meeting at the point A and similarly the amount of current flowing through the wires, meeting at B is also I/4. Thus a current I/2 flows through the conductor AS, i.e. 

I0 = I/2

Hence,  R = R0/2

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