A metal wire of resistance R is cut into three equal pieces...

A metal wire of resistance $R$ is cut into three equal pieces that are then connected side by side to form a new wire the length of which is equal to one-third the original length. What is the resistance of this new wire?

Key Concepts

Resistance and Its Dependence on Geometry

The resistance of a material depends on its intrinsic resistivity as well as its geometric factors, specifically the length of the conductor and its cross-sectional area. The formula R = ?L/A shows that resistance increases with length and decreases with a larger area. When physical dimensions of the conductor change, its resistance changes proportionally, a concept that is critical in analyzing how modifications like cutting or combining wires affect overall resistance.

Cutting and Combining Conductors

When a conductor is cut into pieces, each segment’s resistance changes according to its new length, given that the cross-sectional area and material properties remain constant. In the context of combining these cut segments to form a new configuration, it is essential to understand how these individual resistances contribute to the total resistance, which can vary significantly based on whether they are connected in series or in parallel.

Parallel Resistor Networks

A key concept in combining resistive elements is the parallel configuration, where conductors are connected side by side. In a parallel network, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances. This arrangement reduces the overall resistance compared to each individual resistor because the current has multiple paths to flow. Understanding this concept is crucial when analyzing circuits where identical segments are arranged in parallel, as the net resistance is lower than that of any single segment.

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