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Electric Current

Problem 3.153

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.R_{0} . Find the resistance RR of the whole grid between points AA and BB. Instruction. Make use of principles of symmetry and superposition.

Reveal Answer
3.153. Imagine the voltage VV to be applied across the points AA and B.B . Then V=IR=I0R0,V=I R=I_{0} R_{0}, where II is the current carried by the lead wires, I0I_{0} is the current carried by the conductor ABA B. The current I0I_{0} can be represented as a superposition of two_currents. If the current II flowed into point AA and spread all over] the infinite wire grid, the conductor ABA B would carry (because of symmetry) the current I/4I / 4. Similarly, if the current II flowed into the grid from infinity and left the grid through point B,B, the conductor ABA B would also carry the current I/4I / 4. Superposing both of these solutions, we obtain I0=I/2.I_{0}=I / 2 . Therefore, R=R0/2.RAC=23R0R=R_{0} / 2 . \quad R_{A C}=\frac{2}{3} R_{0}
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