Objective: Calculate the current flowing through the 20 \(\Omega\) resistor in the given circuit.
Circuit Configuration Analysis
The circuit consists of:
- Two resistors (\(R_1 = 20\ \Omega\), \(R_2 = 30\ \Omega\)) connected in parallel.
- This parallel combination is connected in series with another resistor (\(R_3 = 8\ \Omega\)).
- The entire series-parallel combination is connected to a 12-V battery.
Equivalent Resistance Calculation
First, find the equivalent resistance of the parallel resistors (\(R_p\)):
\( R_p = \frac{R_1 \times R_2}{R_1 + R_2} = \frac{20\ \Omega \times 30\ \Omega}{20\ \Omega + 30\ \Omega} = \frac{600\ \Omega^2}{50\ \Omega} = 12\ \Omega \)
Next, find the total equivalent resistance (\(R_{total}\)) of the circuit, considering \(R_p\) is in series with \(R_3\):
\( R_{total} = R_p + R_3 = 12\ \Omega + 8\ \Omega = 20\ \Omega \)
Total Circuit Current
Calculate the total current (\(I_{total}\)) flowing from the battery using Ohm's Law (\(V = I \times R\)):
\( I_{total} = \frac{V_{battery}}{R_{total}} = \frac{12\ \text{V}}{20\ \Omega} = 0.6\ \text{A} \)
Voltage Across Parallel Section
This total current (\(I_{total}\)) flows through the series resistor (\(R_3\)) and then splits between the parallel resistors (\(R_1\) and \(R_2\)). The voltage drop across the parallel section (\(V_p\)) is:
\( V_p = I_{total} \times R_p = 0.6\ \text{A} \times 12\ \Omega = 7.2\ \text{V} \)
Current Through 20 \(\Omega\) Resistor
The voltage across the parallel combination (\(V_p\)) is the same for both the \(20\ \Omega\) and \(30\ \Omega\) resistors. Apply Ohm's Law to find the current (\(I_{20\ \Omega}\)) through the \(20\ \Omega\) resistor (\(R_1\)):
\( I_{20\ \Omega} = \frac{V_p}{R_1} = \frac{7.2\ \text{V}}{20\ \Omega} = 0.36\ \text{A} \)
Therefore, 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 :
