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?

Verified step by step guidance

1. Identify that the wire is cut into three equal parts, each with resistance R/3.
2. Recognize that one of these parts is formed into a circle, which is then connected between the other two parts.
3. Note that the circle can be considered as two resistors of R/6 each in parallel.
4. Calculate the equivalent resistance of the two R/6 resistors in parallel using the formula: 1/R_eq = 1/(R/6) + 1/(R/6).
5. Combine the equivalent resistance of the circle with the two R/3 resistors in series to find the total resistance between points a and b.

Key Concepts

Resistance in Series and Parallel Circuits

In electrical circuits, resistors can be arranged in series or parallel configurations. In a series circuit, the total resistance is the sum of individual resistances, while in a parallel circuit, the total resistance can be calculated using the formula 1/R_total = 1/R1 + 1/R2 + ... + 1/Rn. Understanding these configurations is crucial for analyzing the overall resistance in complex circuits.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed as V = IR. It is fundamental in circuit analysis, allowing us to relate voltage, current, and resistance.

Equivalent Resistance

Equivalent resistance is the total resistance of a circuit when multiple resistors are combined. For resistors in series, the equivalent resistance is simply the sum of the individual resistances. For resistors in parallel, the equivalent resistance is found using the reciprocal formula. This concept is essential for simplifying complex circuits to determine the overall resistance between two points.