Problem 49 - College Physics

Question

A 40.0\(\Omega\) resistor and a 90.0\(\Omega\) resistor are connected in parallel, and the combination is connected across a \(120-\mathrm{V}\) dc line. (a) What is the resistance of the parallel combination? (b) What is the total current through the parallel combination? (c) What is the current through each resistor?

Step-by-Step Solution

Answer

(a) 27.7 Ω, (b) 4.33 A, (c) 3.0 A for 40 Ω and 1.33 A for 90 Ω.

Step 1: Understanding Parallel Resistors

When resistors are connected in parallel, the reciprocal of the equivalent resistance is the sum of the reciprocals of each individual resistance. Mathematically, this is given by \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \) where \( R_1 = 40.0 \Omega \) and \( R_2 = 90.0 \Omega \).

Step 2: Calculating Equivalent Resistance

Apply the formula for parallel resistors to find the equivalent resistance:\[\frac{1}{R_{eq}} = \frac{1}{40.0} + \frac{1}{90.0} = \frac{1}{40.0} + \frac{1}{90.0}\]Calculating this gives:\[\frac{1}{R_{eq}} = 0.025 + 0.0111 = 0.0361\]Thus, \( R_{eq} = \frac{1}{0.0361} \approx 27.7 \Omega \).

Step 3: Calculating Total Current Through the Parallel Combination

Use Ohm's law \( I = \frac{V}{R} \) to find the total current. The voltage across the resistors is given as \( 120 \mathrm{V} \), and the equivalent resistance is \( 27.7 \Omega \). Substituting these values gives:\[I_{total} = \frac{120}{27.7} \approx 4.33 \mathrm{A}\]

Step 4: Calculating Current Through Each Resistor

For resistors in parallel, the voltage across each resistor is the same (\( 120 \mathrm{V} \)). Use Ohm's law to find the current through each:1. For the \( 40.0 \Omega \) resistor: \[ I_1 = \frac{120}{40.0} = 3.0 \mathrm{A} \]2. For the \( 90.0 \Omega \) resistor: \[ I_2 = \frac{120}{90.0} \approx 1.33 \mathrm{A} \]

Key Concepts

Ohm's Law

Ohm's Law is one of the fundamental principles used to solve electrical circuits. It states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance between them. The mathematical expression is given by:

\[ I = \frac{V}{R} \]where:

• \( I \) is the current in amperes (A),
• \( V \) is the voltage in volts (V),
• \( R \) is the resistance in ohms (\( \Omega \)).

In the exercise, this law helps calculate the current passing through each resistor in the circuit. By knowing the voltage and the resistance, you can easily determine how much current flows. Remember, this concept applies universally to both series and parallel circuits.

Equivalent Resistance

The concept of equivalent resistance is key when dealing with circuits that involve multiple resistors. In parallel circuits, resistors share voltage, but divide the total current among them. The equivalent resistance in a parallel circuit can be calculated using:

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n} \]which simplifies to:

\[ R_{eq} = \frac{1}{\left( \frac{1}{R_1} + \frac{1}{R_2} \right)} \]
For the provided example, combining a 40\( \Omega \) and a 90\( \Omega \) resistor results in an equivalent resistance of approximately 27.7\( \Omega \). Calculating equivalent resistance helps you simplify complex circuits into single manageable resistances, which then allows easier analysis of the whole system.

Current Calculation

Calculating current in electrical circuits involves using both Ohm’s Law and the concept of equivalent resistance. In any circuit, once you have calculated the equivalent resistance for a parallel setup, the total current flowing through the circuit can be easily determined using:

\[ I_{total} = \frac{V}{R_{eq}} \]
Using the exercise as a reference, with an equivalent resistance of 27.7\( \Omega \) and a power supply of 120V, the total current through the circuit is approximately 4.33A.

For individual resistors in a parallel configuration, the current can be calculated separately by using the voltage across the resistors:

• For the 40\( \Omega \) resistor, the current is 3.0A.
• For the 90\( \Omega \) resistor, the current is around 1.33A.

This breakdown provides insight into how current is distributed amongst the components connected in parallel.