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 \(0.36\ A\).
Important Questions from Combination of Resistors — Series and Parallel
Consider two resistors, $R_1$ and $R_2$, connected in series to a DC voltage source. Which of the following statements accurately describes the distribution of current and voltage across these resistors?
A cell of negligible resistance and e.m.f 2 volt is connected to series combination of 2 ohm, 3 ohm and 5 ohm. The potential difference across the 3 ohm resistance is:
Two bulbs A, of (100w, 100v), and B of (60 w, 100v) are connected in series and across the series combination 200 v is applied. Which bulb will be fused?
3 resistors of 3 ohm each connected in series. What is the mean values of resistors?
The equivalent resistance of the resistances (two) joined in parallel is 6/5 Ω. When one of the resistance wire is broken, the effective resistance becomes 2Ω. The resistance of the wire that got broken was :
