Each of two identical uniform metal bars has a resistance R...

Each of two identical uniform metal bars has a resistance $R .$ If they are welded together along one-third of their lengths (see Figure $19.54 ),$ what is the resistance of this combination in terms of $R ?$

Key Concepts

Circuit Simplification Using Symmetry

Exploiting symmetry in a circuit can significantly simplify analysis by revealing equivalent resistor networks. Recognizing symmetrical arrangements helps in identifying series and parallel portions, thus reducing complex combinations into simpler equivalent networks that are easier to analyze.

Parallel Resistor Combination

In a parallel arrangement, the same voltage is applied across each resistor, and the total conductance is the sum of the individual conductances. The overall resistance of parallel resistors is found by taking the reciprocal of the sum of reciprocals of the individual resistances, a key method when voltage divides between different conduction paths.

Uniform Conductor Resistivity

In a uniform conductor, the intrinsic resistance is determined by its material properties and geometry. Specifically, the resistance is directly proportional to the length and inversely proportional to the cross-sectional area. This principle allows one to subdivide a uniform resistance into parts corresponding to fractional lengths of the conductor.

Series Resistor Combination

When resistors are connected in series, they carry the same current and their resistances add directly to yield the total resistance of the connection. This concept is vital when analyzing segments of a continuous conductor or when combining serial elements in a circuit.

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