Problem 108 - Physics

Question

Multiple-Concept Example 9 discusses the physics principles used in this problem. Three resistors, \(2.0,4.0\), and \(6.0 \Omega\), are connected in series across a \(24-V\) battery. Find the power delivered to each resistor.

Step-by-Step Solution

Answer

The power delivered to the 2.0 Ω resistor is 8.0 W, to the 4.0 Ω resistor is 16.0 W, and to the 6.0 Ω resistor is 24.0 W.

Step 1: Calculate the Total Resistance

Since the resistors are connected in series, the total resistance \( R_{total} \) is the sum of the individual resistances:\[R_{total} = R_1 + R_2 + R_3 = 2.0 \, \Omega + 4.0 \, \Omega + 6.0 \, \Omega = 12.0 \, \Omega\]

Step 2: Find the Total Current

Using Ohm's Law, calculate the total current \( I \) flowing through the circuit:\[I = \frac{V}{R_{total}} = \frac{24 \, V}{12.0 \, \Omega} = 2.0 \, A\]

Step 3: Calculate Power Across Each Resistor

The power delivered to a resistor in a series circuit can be found using the formula \( P = I^2 R \). For the 2.0 \( \Omega \) resistor:\[P_1 = I^2 \times R_1 = (2.0 \, A)^2 \times 2.0 \, \Omega = 8.0 \, W\]For the 4.0 \( \Omega \) resistor:\[P_2 = I^2 \times R_2 = (2.0 \, A)^2 \times 4.0 \, \Omega = 16.0 \, W\]For the 6.0 \( \Omega \) resistor:\[P_3 = I^2 \times R_3 = (2.0 \, A)^2 \times 6.0 \, \Omega = 24.0 \, W\]

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in the study of electrical circuits that relates the voltage, current, and resistance within a circuit. It is mathematically expressed as \( V = I \times R \), where:

• \( V \) stands for voltage, measured in volts (V),
• \( I \) stands for current, measured in amperes (A),
• \( R \) stands for resistance, measured in ohms (Ω).

This equation shows that voltage (\( V \)) across a resistor is directly proportional to the current (\( I \)) flowing through it, multiplied by the resistance (\( R \)) of the resistor.
In practical terms, Ohm's Law helps us determine how much current will flow through a circuit for a given voltage and resistance. It can be applied to find either of the three parameters if the other two are known.
For example, in the exercise with a 24-V battery across resistors in series, knowing the total resistance allows us to calculate the current using Ohm's Law, as was demonstrated in the step-by-step solution.

Total Resistance Calculation

To find the total resistance in a circuit where resistors are connected in series, you simply add up the resistance values of all resistors.
The formula for the total resistance, \( R_{total} \), in a series circuit is:

• \( R_{total} = R_1 + R_2 + R_3 + \ldots + R_n \)

For example, if you have resistors of 2.0 Ω, 4.0 Ω, and 6.0 Ω connected in series, their total resistance would be:\[ R_{total} = 2.0 \, \Omega + 4.0 \, \Omega + 6.0 \, \Omega = 12.0 \, \Omega \]
In a series circuit, each resistor is connected end-to-end, and the same current flows through each one. Therefore, calculating the total resistance this way is valuable because it tells us how much the resistors collectively resist the flow of electricity. Understanding this concept is crucial for designing circuits and ensuring they function correctly within specific voltage and current limits.

Power Calculation in Circuits

Power in electrical circuits is a measure of the rate at which energy is used, absorbed, or produced by a component. It is defined by the equation:\[ P = I^2 \times R \] where:

• \( P \) is power in watts (W),
• \( I \) is the current in amperes (A),
• \( R \) is the resistance in ohms (Ω).

With this formula, you can compute the power delivered to each resistor separately in a series circuit.
In our exercise, we found that the power delivered to resistors of 2.0 Ω, 4.0 Ω, and 6.0 Ω is 8.0 W, 16.0 W, and 24.0 W respectively.
Each resistor, when connected in series to the same current, will have a distinct power based on its resistance. This concept is particularly important for understanding energy consumption and ensuring that circuit components operate within safe power limits to avoid damage or failure. Understanding power calculations is key in assessing how much energy different parts of a circuit are using, helping with the design and analysis of electrical systems.