A uniform wire of resistance R is cut into three equal lengths. One of these is formed into a circle and connected between the other two (Fig. E26.1). What is the resistance between the opposite ends a and b?

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First, understand the configuration of the circuit. The wire of resistance R is divided into three equal parts, each with resistance R/3. One part is formed into a circle, and the other two parts are connected in series with the circle.

Identify the resistances in the circuit. The circle formed by one part of the wire has a resistance of R/3. The other two parts, each with resistance R/3, are connected in series with the circle.

Calculate the equivalent resistance of the circle. Since the circle is connected between two points, it acts as a parallel resistor to the other two resistors. The equivalent resistance of the circle is R/3.

Determine the total resistance between points a and b. The two resistors in series (each R/3) have a combined resistance of 2R/3. The circle (R/3) is in parallel with this series combination.

Use the formula for parallel resistors to find the total resistance: \( R_{total} = \frac{1}{\frac{1}{R_{series}} + \frac{1}{R_{circle}}} \), where \( R_{series} = 2R/3 \) and \( R_{circle} = R/3 \). Calculate the equivalent resistance using this formula.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Resistance in Series and Parallel

In a circuit, resistors can be arranged in series or parallel, affecting the total resistance. For resistors in series, the total resistance is the sum of individual resistances. In parallel, the reciprocal of the total resistance is the sum of the reciprocals of individual resistances. Understanding these arrangements is crucial for calculating the equivalent resistance in complex circuits.

Equivalent Resistance

Equivalent resistance is the single resistance that can replace a combination of resistors without changing the current or voltage in the circuit. It simplifies the analysis of circuits by reducing multiple resistors to a single value. Calculating equivalent resistance involves identifying series and parallel connections and applying the appropriate formulas.

Symmetry in Circuits

Symmetry in circuits can simplify the analysis by allowing certain assumptions about current distribution. If a circuit is symmetric, components may share equal current or voltage, reducing complexity. Recognizing symmetry can help identify points where potential differences are zero, aiding in the calculation of equivalent resistance.