Circuit Analysis: Parallel and Series Resistors

This problem involves calculating the current through a specific resistor in a mixed series-parallel circuit. We need to determine the equivalent resistance and voltage distribution.

Calculating Equivalent Parallel Resistance

First, find the equivalent resistance (\(R_p\)) of the \(20\ \Omega\) and \(30\ \Omega\) resistors connected in parallel:

Using the formula for parallel resistors: \(R_p = \frac{R_1 \times R_2}{R_1 + R_2}\)

Substitute the values: \(R_p = \frac{20\ \Omega \times 30\ \Omega}{20\ \Omega + 30\ \Omega} = \frac{600\ \Omega^2}{50\ \Omega} = 12\ \Omega\)

Determining Total Circuit Resistance

The parallel combination (\(R_p = 12\ \Omega\)) is connected in series with an \(8\ \Omega\) resistor. Calculate the total equivalent resistance (\(R_{total}\)):

\(R_{total} = R_p + R_{series}\)

\(R_{total} = 12\ \Omega + 8\ \Omega = 20\ \Omega\)

Finding Total Current from Battery

Using Ohm's Law (\(V = IR\)), calculate the total current (\(I_{total}\)) flowing from the 12-V battery:

\(I_{total} = \frac{V_{battery}}{R_{total}}\)

\(I_{total} = \frac{12\ V}{20\ \Omega} = 0.6\ A\)

Calculating Voltage Across Parallel Section

This total current flows through the parallel combination. Calculate the voltage drop (\(V_p\)) across the parallel resistors:

\(V_p = I_{total} \times R_p\)

\(V_p = 0.6\ A \times 12\ \Omega = 7.2\ V\)

Calculating Current Through 20 Ohm Resistor

The voltage across the parallel combination (\(V_p\)) is the same for both the \(20\ \Omega\) and \(30\ \Omega\) resistors. Now, find the current (\(I_{20}\)) through the \(20\ \Omega\) resistor using Ohm's Law:

\(I_{20} = \frac{V_p}{R_{20}}\)

\(I_{20} = \frac{7.2\ V}{20\ \Omega} = 0.36\ A\)

The current through the \(20\ \Omega\) resistor is \(0.36\ A\).