The circuit to the right is constructed using 9 identical batteries and 9 identical resistors. the lower left corner of the circuit is grounded. determine whether the current through each resistor is zero or non-zero.

The R1, R2, R3, R5 and R6 contains current and R4, R7, R8 and R9 do not contain current. Since R8 loop voltages sources cancel VR8 + V – V = 0 so VR8 = 0. The top of R9 potential from ground is +V + V + V – V -  V = V. The bottom of R9 potentials from ground is V so VR9 = 0. The left of R7 potential from ground is +V + V – V = V. The right side of R7 potential from ground is +V + V +V – V – V = V. Therefore, VR7 = 0. From the ground the left side of R4 = +V and from the ground the right side or R4 is at V so VR4 = 0. Since we know that R1 must have 2V across it and we know that R6 and R3 each have V/2, R2 has V across it and R5 must have V across it because the potential of its left end from the ground = +V + V = 2V but the potential of its right end from the ground must be +V + V + V – V – V = V. 

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