A symmetric circuit

Electricity and Magnetism Level 3

In the diagram below, E = 12 V E = 12 V and all the resistors have equal resistances. Calculate the voltages V A B = V A − V B , V A C = V A − V C , V_{AB}=V_A - V_B, V_{AC}= V_A - V_C, and V A D = V A − V D V_{AD} = V_A - V_D in volts and input your answer as their sum.


The answer is 4.

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Djordje Veljkovic by Djordje Veljkovic , 19, Serbia

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1 solution

Steven Chase
Jun 6, 2017

Relevant wiki: Simple Circuits

Because of symmetry, the horizontal resistors tying the left and right sides must have equal potentials on both sides. They therefore have no current flowing through them, and can be treated as open circuits. See the revised diagram. This effectively splits the circuit into two identical isolated halves. The voltages at A and B can be found through voltage division.

U a b = 8 − 8 = 0 U a c = 8 − 12 = − 4 U a d = 8 − 0 = 8 U a b + U a c + U a d = 0 − 4 + 8 = 4 \large{U_{ab} = 8 - 8 = 0 \\ U_{ac} = 8 - 12 = -4 \\ U_{ad} = 8 - 0 = 8 \\ U_{ab} + U_{ac} + U_{ad} = 0 - 4 + 8 = \boxed{4}}

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