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Symmetry in Resistance Udit Gupta

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Resistor Shapes l Three identical 1 resistors are connected in parallel as shown below. What is the total resistance of the circuit? A, 3 B.1 C. 1/3 D. 312 E. 2/3 +1V Ground, or 0 V at that point

Solution Answer: C Justification: This is an example of three identical resistors connected in parallel. We know that the equation for parallel resistors is: otal R otal This is a straightforward approach to solving this problem. However, we can also consider the symmetry of the problem and Ohm's law to take a different more elegant approach. It will be discussed in the next slide

Solution - Continued Answer: C Justification: We can use the symmetry arguments to solve this problem. This argument will be especially useful for solving more difficult problems. In this case, the three resistors are equal, therefore, the branches have equal resistances and the current through each one of them must be equal: V IR R Total Total Total of braches Notice, in a parallel circuit that has identical branches the total resistance is LESS than the resistance of each one of the branches. The current through each is one of them equals 1/3 of the total current equal currents flow through each one of the identical parallel branches

Resistor Shapes ll identical 1 resistors are arranged like the image below. What is the total resistance of the circuit? A, 1 B. 2 C. 1/2 D. 1/3 E. 1/4 +1V LGround, or 0 V at that point

Solution Answer: A Justification: All the resistors are identical, so using the symmetry of the circuit we find that the current splits in half and passes through two resistors. The center resistor is ignored because the ends of the resistor have the same potential, as the circuit is symmetric. In other words, the potential of points A and B will be the same (no current flow from A to B or from B to A). Therefore, you can ignore segment AB. Then you have two identical 2-Ohm branches connected in parallel: Rbranch 1 +1

Resistor Shapes llI identical 1 resistors are arranged in a tetrahedron. What is the total resistance of the circuit? A, 6 B. 5 C.2 D.1 E. 1/2 .1V LGround, or 0 V at that point

Solution Answer: E Justification: The left and right side of the circuit is symmetric, and therefore no current flows through the bottom resistor (AB) because the ends have the same potential. We can find Rtotal by considering the 2 resistors in series on the left and right, and the central resistor. .tV 2 R,lal (1+1) (1+1) Rola, 1(2 total The total resistance is less than the resistance of each one of the branches.

Solution Answer: C Justification: The cube here is highly symmetric. We chose any possible path for the current to go from point A to B. If we start at the current source (A), we see that the current divides into thirds, then into sixths, and the combines back into thirds. This gives us: V-Va any paralldlbramnch RRRI t63 6 5 5 /3 V 5RI 5 I 6 6 Total_