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27. Resistors & DC Circuits

Solving Resistor Circuits

Physics

27. Resistors & DC Circuits

Solving Resistor Circuits

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3:54 minutes

Problem 6b

Textbook Question

Three 45-Ω lightbulbs and three 65-Ω lightbulbs are connected in series. What is the total resistance if all six are wired in parallel?

Verified step by step guidance

Identify the given resistances: There are three 45-Ω resistors and three 65-Ω resistors. Since the problem asks for the total resistance when all six resistors are connected in parallel, we will use the formula for the equivalent resistance of resistors in parallel.

Recall the formula for the equivalent resistance of resistors in parallel:

R_{eq} = \frac{1}{1 / R_{1} + 1 / R_{2} + \dots}

Substitute the resistances into the formula: Since there are three 45-Ω resistors and three 65-Ω resistors, the total equivalent resistance can be calculated as:

R_{eq} = \frac{1}{1 / 45 + 1 / 45 + 1 / 45 + 1 / 65 + 1 / 65 + 1 / 65}

Simplify the terms in the denominator: Combine the terms for the three 45-Ω resistors and the three 65-Ω resistors separately. This gives:

R_{eq} = \frac{1}{3 / 45 + 3 / 65}

Combine the fractions and calculate the reciprocal to find the total equivalent resistance. This will give you the final value of the total resistance for all six resistors connected in parallel.

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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 Circuits

In electrical circuits, resistors can be connected in series or parallel, affecting the total resistance. In a series connection, the total resistance is the sum of individual resistances, while in a parallel connection, the total resistance can be calculated using the formula 1/R_total = 1/R1 + 1/R2 + ... + 1/Rn, where R1, R2, etc., are the resistances of the individual components.

Ohm's Law

Ohm's Law is a fundamental principle in electronics that relates voltage (V), current (I), and resistance (R) in a circuit, expressed as V = I × R. This law is essential for understanding how voltage drops across resistors and how current flows through different configurations of resistors, impacting the overall circuit behavior.

Equivalent Resistance

Equivalent resistance is a single resistance value that can replace a combination of resistors in a circuit without changing the current or voltage characteristics. For parallel circuits, the equivalent resistance is always less than the smallest individual resistance, which is crucial for calculating the total resistance when multiple resistors are connected in parallel.

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A 10.0-m length of wire consists of 5.0 m of copper followed by 5.0 m of aluminum, both of diameter 1.4 mm. A voltage difference of 75 mV is placed across the composite wire. (a) What is the total resistance (sum) of the two wires? (b) What is the current through the wire? (c) What are the voltages across the aluminum part and across the copper part?

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Textbook Question

A 2.5-kΩ and a 3.7-kΩ resistor are connected in parallel; this combination is connected in series with a 1.4-kΩ resistor. If each resistor is rated at 1.0 W (maximum without overheating), what is the maximum voltage that can be applied across the whole network?

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Textbook Question

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