Two of the three resistors in FIGURE EX28.23 are unknown but equal. The total resistance between points 1 and 2 is 75Ω. What is the value of R?

검증된 단계별 안내

Step 1: Analyze the circuit configuration. The circuit consists of three resistors: two resistors with unknown resistance R and one resistor with a resistance of 200Ω. The two resistors R are in parallel with each other, and this parallel combination is in series with the 200Ω resistor.

Step 2: Calculate the equivalent resistance of the two parallel resistors R. The formula for the equivalent resistance of two resistors in parallel is:

R_{eq} = \frac{1}{\frac{1}{R} + \frac{1}{R}}

. Simplify this expression to find the equivalent resistance of the parallel combination.

Step 3: Add the equivalent resistance of the parallel combination to the resistance of the 200Ω resistor, since the parallel combination is in series with the 200Ω resistor. The total resistance of resistors in series is given by:

R_{total} = R_{eq} + 200

Step 4: Set the total resistance equal to the given value of 75Ω. This gives the equation:

75 = \frac{1}{\frac{1}{R} + \frac{1}{R}} + 200

. Solve this equation for R.

Step 5: Rearrange and simplify the equation to isolate R. Use algebraic manipulation to solve for R, ensuring all terms are correctly handled. This will yield the value of R.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 current, voltage, or resistance.

Series and Parallel Resistors

Resistors can be arranged in series or parallel configurations, affecting the total resistance in a circuit. In a series configuration, the total resistance is the sum of individual resistances (R_total = R1 + R2 + ...). In a parallel configuration, the total resistance can be calculated using the formula 1/R_total = 1/R1 + 1/R2 + ... This distinction is essential for solving problems involving multiple resistors, as seen in the given circuit.

Equivalent Resistance

Equivalent resistance is the total resistance of a circuit that can replace a combination of resistors without changing the current or voltage in the circuit. For resistors in parallel, the equivalent resistance is always less than the smallest individual resistor. In the context of the question, calculating the equivalent resistance between points 1 and 2 is necessary to find the unknown resistor value (R) when the total resistance is given.