A machine part has a resistor X protruding from an opening in the side. This resistor is connected to three other resistors, as shown in Fig. E26.2. An ohmmeter connected across a and b reads 2.00 Ω.
What is the resistance of X?

Verified step by step guidance

1. Identify the resistors in the circuit and their configuration. The resistors are 14.0 Ω, 13.0 Ω, 15.0 Ω, and the unknown resistor X.
2. Recognize that the 14.0 Ω and 13.0 Ω resistors are in parallel. Calculate their equivalent resistance using the formula: \( \frac{1}{R_{eq1}} = \frac{1}{14.0} + \frac{1}{13.0} \).
3. Combine the equivalent resistance of the 14.0 Ω and 13.0 Ω resistors with the 15.0 Ω resistor in series. The total resistance for this part of the circuit is: \( R_{total1} = R_{eq1} + 15.0 \).
4. Recognize that the total resistance of the circuit is given as 2.00 Ω. This total resistance is the combination of the unknown resistor X in parallel with the previously calculated total resistance: \( \frac{1}{R_{total}} = \frac{1}{R_{total1}} + \frac{1}{X} \).
5. Solve for the unknown resistor X using the equation: \( \frac{1}{2.00} = \frac{1}{R_{total1}} + \frac{1}{X} \).

Key Concepts

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 mathematically as V = IR. Understanding this law is crucial for analyzing electrical circuits and calculating unknown values such as resistance.

Series and Parallel Resistor Configurations

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

Equivalent Resistance

Equivalent resistance is the total resistance of a circuit or a portion of a circuit that can replace multiple resistors. It simplifies circuit analysis by allowing us to treat complex arrangements of resistors as a single resistor. Calculating the equivalent resistance is vital for solving problems involving multiple resistors, especially when determining the resistance across specific points in a circuit.