Problem 41 - Physics

Question

A \(36.0-\Omega\) resistor and an \(18.0-\Omega\) resistor are connected in series across a \(15.0-\) battery. What is the voltage across (a) the \(36.0-\Omega\) resistor and (b) the \(18.0-\Omega\) resistor?

Step-by-Step Solution

Answer

(a) 10.0 V, (b) 5.0 V.

Step 1: Understand Series Resistor Circuit

In a series circuit, the total resistance is the sum of the individual resistances. Here, we have a 36 \( \Omega \) and an 18 \( \Omega \) resistor connected in series across a 15 V battery.

Step 2: Calculate Total Resistance

Add the resistances of the two resistors in series to find the total resistance: \( R_{total} = 36 \ \Omega + 18 \ \Omega = 54 \ \Omega \).

Step 3: Apply Ohm’s Law to Find Total Current

Ohm’s Law states \( V = IR \). Using the total resistance and voltage, the total current \( I \) in the circuit is \( I = \frac{V}{R_{total}} = \frac{15.0 \ \mathrm{V}}{54 \ \Omega} \approx 0.278 \ \mathrm{A} \).

Step 4: Find Voltage Across 36.0-Ω Resistor

The voltage across a resistor in a series circuit can be found using Ohm’s Law: \( V = IR \). For the 36.0-Ω resistor, \( V = 0.278 \ \mathrm{A} \times 36 \ \Omega = 10.0 \ \mathrm{V} \).

Step 5: Find Voltage Across 18.0-Ω Resistor

Similarly, for the 18.0-Ω resistor, use \( V = IR \): \( V = 0.278 \ \mathrm{A} \times 18 \ \Omega = 5.0 \ \mathrm{V} \).

Key Concepts

Series Circuit

In a series circuit, all the components are connected one after another, forming a single path for the current to flow. This means that the same current flows through each component, including resistors.
When resistors are connected in series, their resistances add up to form the total resistance of the circuit. This can be calculated as:

• \( R_{\text{total}} = R_1 + R_2 + \ldots + R_n \)

In our example, a \(36.0-\Omega\) resistor and an \(18.0-\Omega\) resistor are in series, so their total resistance is \(54.0-\Omega\).
This principle of adding resistances in series helps us understand how the circuit's total resistance affects the current flow, governed by Ohm's Law. Understanding the series circuit configuration is crucial in determining how voltage divides across each component.

Resistor Voltage

The voltage across each resistor in a series circuit depends on its resistance value and the total current passing through the circuit. According to Ohm’s Law, the voltage (\( V \)) across a resistor is the product of the current (\( I \)) through it and its resistance (\( R \)), expressed as:

• \( V = IR \)

After we calculated the total current in the circuit (\(0.278\, \mathrm{A}\)), we applied this formula to find the voltage across each resistor:
* For the \(36.0-\Omega\) resistor, the voltage is \(10.0\, \mathrm{V}\).
* For the \(18.0-\Omega\) resistor, it is \(5.0\, \mathrm{V}\).
Notice how the sum of these voltages equals the total voltage of the battery \(15.0\, \mathrm{V}\). This is a key property of series circuits: the total voltage supplied is divided among the resistors according to their resistance values. This division of voltage is also known as "voltage drop." It helps allocate electrical energy across different parts of the circuit, based on their resistive properties.

Circuit Analysis

Circuit analysis involves methods and laws to determine the various electrical parameters such as current, voltage, and resistance within an electrical circuit. It's an essential skill for understanding and designing electronic pathways.
Applying Ohm’s Law, which states \( V = IR \), is a fundamental step in circuit analysis. By knowing any two of these variables in a circuit, we can calculate the third. In this exercise:

• The total resistance was calculated by simply adding the resistors in series.
• The total current was found using this total resistance and the battery's voltage.
• Finally, the voltage across each resistor was determined using the current and individual resistances.

This step-by-step approach ensures accuracy and provides a clear roadmap for solving more complex circuit problems.
In practice, applying such systematic analysis enables you to predict how circuits will behave. This can assist in troubleshooting issues or designing circuits to meet specific needs. Mastering these concepts can significantly deepen your understanding of electronics and electrical systems.