Question

Two resistors with resistances \(200 . \Omega\) and \(400 . \Omega\) are connected (a) in series and (b) in parallel with an ideal 9.00-V battery. Compare the power delivered to the \(200 .-\Omega\) resistor.

Short answer

Answer: The power delivered to the \(200\Omega\) resistor when connected in series is \(0.045W\), and the power delivered when connected in parallel is \(0.405W\).

Step-by-step solution

Find the Total Resistance in Series Connection

For resistors in series, the total resistance is the sum of the individual resistances. So, \(R_{total} = R_1 + R_2 = 200\Omega + 400\Omega = 600\Omega\).

Find the Current in Series Connection

Use Ohm's law, \(V = IR\), to find the current flowing through the circuit. Rearranging the equation and solving for the current gives \(I = \frac{V}{R_{total}} = \frac{9V}{600\Omega} = 0.015A\).

Calculate the Power Delivered to the \(200\Omega\) Resistor in Series Connection

We can now calculate the power delivered to the \(200\Omega\) resistor using the formula \(P = I^2R\). For the series connection, the current through both resistors is the same. Thus, \(P_{series} = (0.015A)^2(200\Omega) = 0.045W\).

Find the Equivalent Resistance in Parallel Connection

For resistors in parallel, the equivalent resistance is given by the formula \(\frac{1}{R_{equiv}} = \frac{1}{R_1} + \frac{1}{R_2}\). Plugging in the values, we get \(\frac{1}{R_{equiv}} = \frac{1}{200\Omega} + \frac{1}{400\Omega}\), which gives us \(R_{equiv} = 133.33\Omega\).

Find the Total Current in Parallel Connection

Again, use Ohm's law to find the total current flowing through the circuit. In this case, \(I_{total} = \frac{V}{R_{equiv}} = \frac{9V}{133.33\Omega} = 0.0675A\).

Find the Current through the \(200\Omega\) Resistor in Parallel Connection

We can find the current through the \(200\Omega\) resistor in a parallel connection using the current divider rule, which is as follows: \(I_{200} = I_{total}\frac{R_2}{R_1 + R_2} = 0.0675A \frac{400\Omega}{200\Omega +400\Omega} = 0.045A\).

Calculate the Power Delivered to the \(200\Omega\) Resistor in Parallel Connection

Now, we can calculate the power delivered to the \(200\Omega\) resistor using the formula \(P = I^2R\). For the parallel connection, \(P_{parallel} = (0.045A)^2(200\Omega) = 0.405W\). So, the power delivered to the \(200\Omega\) resistor when connected in series with the \(400\Omega\) resistor is \(0.045W\), and the power delivered when connected in parallel is \(0.405W\).

Key concepts

Series and Parallel Circuits

When dealing with electrical circuits, especially those involving resistors, it's important to understand the difference between series and parallel configurations.

In a **series circuit**, resistors are arranged end to end, meaning the current has only one path to follow. Because of this, the total resistance in a series circuit is simply the sum of all resistances:

• The total resistance formula is: \( R_{total} = R_1 + R_2 + ... + R_n \).
• The current remains the same through each component.
• This configuration can result in higher total resistance, which decreases the total current flow under constant voltage.

On the other hand, **parallel circuits** allow multiple pathways for the current to flow. Here, each resistor has its own separate branch connecting to both ends of the power source.

In a parallel configuration:

• The reciprocal of the total resistance equals the sum of the reciprocals of each individual resistance: \( \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n} \).
• Voltage across each resistor is the same.
• Different branches can carry different currents depending on their resistance.

Both configurations impact the total resistance and the behavior of the current in dissimilar ways. Understanding these fundamental differences is key for predicting how a circuit will function when resistors are added or rearranged.

Power in Resistors

Power in resistors can tell us how much energy is being used in a circuit. It depends on both the current flowing through the resistor and the resistance itself. Understanding how to calculate power is important for assessing the performance and safety of electrical circuits.

The power absorbed by a resistor can be calculated using one of the following formulas, depending on the available data:

• Using current and resistance: \( P = I^2 R \) , where **I** is current and **R** is resistance.
• Using voltage and resistance: \( P = \frac{V^2}{R} \), where **V** is the voltage across the resistor.
• Using voltage and current: \( P = VI \).

In the exercise above, when resistors were connected in series, the current through the 200Ω resistor was lower, leading to less power being consumed compared to when the resistors were in parallel. This illustrates how different circuit arrangements can drastically affect the electrical power used by components within them. Monitoring power is crucial in ensuring devices don’t overheat and damage due to excessive power consumption.

Current Divider Rule

The current divider rule is a principle used primarily in parallel circuits to determine the current flowing through individual resistors. This rule becomes quite useful when multiple resistors are present, and you need to know the distribution of current among them.

In a parallel circuit, the current divider formula helps in calculating the current through each individual branch. The current through a resistor:

• In a two-resistor scenario, is calculated as \( I_{R1} = I_{total} \frac{R_2}{R_1 + R_2} \)
• The idea is that the current in any specific branch is inversely proportional to the resistance of that branch.
• For larger resistance, the path allows less current and vice versa.

In our exercise, the current through the 200Ω resistor when configured in parallel was higher compared to the same resistor in series configuration. This is due to the current divider's ability to distribute the overall current among various branches based on resistance, significantly impacting power distribution and efficiency in a circuit. This rule supports more detailed analysis and predictions about circuit behavior.