Question

series and parallel circuit

1. what is the total resistance of the parallel portion of the circuit?
2. what is the total resistance of the circuit?
3. on the back of the page, draw the simplest equivalent circuit.
4. what is the current at a1?
5. what is the current thru a3 the 10 ohm resistor?
6. what is the current thru a2 the 10 ohm resistor?
7. what is the voltage drop v1 across the 5 ohm resistor?
8. what is the voltage drop v2 across the 10 ohm resistor?
9. what is the voltage drop v3 across the parallel resistor?
10. calculate the remaining voltage at b
11. what is the power used by the 5 ohm resistor?

Explanation:

Step1: Calculate total resistance of parallel portion

For two \(10\ \Omega\) resistors in parallel, use formula \(\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}\). Here \(R_1 = R_2=10\ \Omega\), so \(\frac{1}{R_{eq}}=\frac{1}{10}+\frac{1}{10}=\frac{2}{10}\), then \(R_{eq} = 5\ \Omega\)

Step2: Calculate total resistance of circuit

The circuit has \(5\ \Omega\), \(10\ \Omega\) in series with parallel equivalent \(5\ \Omega\). So \(R_{total}=5 + 10+5=20\ \Omega\)

Step3: Use Ohm's law \(I=\frac{V}{R}\) for current at \(A1\)

Given \(V = 120\ V\) and \(R_{total}=20\ \Omega\), then \(I=\frac{120}{20}=6\ A\)

Step4: For current through \(A3\) (in series - parallel circuit, current divides in parallel. But since parallel resistors are equal (\(10\ \Omega\) each), current from main line (\(6\ A\)) splits equally. So \(I_{A3}=\frac{6}{2}=3\ A\)

Step5: For current through \(A2\) (same as \(A3\) because of equal parallel resistors), \(I_{A2}=3\ A\)

Step6: Voltage drop \(V1\) across \(5\ \Omega\) resistor

Using \(V = IR\), \(I = 6\ A\), \(R = 5\ \Omega\), so \(V1=6\times5 = 30\ V\)

Step7: Voltage drop \(V2\) across \(10\ \Omega\) resistor

\(V2=6\times10=60\ V\)

Step8: Voltage drop \(V3\) across parallel resistor

\(V3=6\times5 = 30\ V\)

Step9: Remaining voltage at \(B\) (same as \(V3\)) is \(30\ V\)

Step10: Power used by \(5\ \Omega\) resistor

Using \(P=I^{2}R\), \(I = 6\ A\), \(R = 5\ \Omega\), so \(P=6^{2}\times5= 180\ W\)

Answer:

1. \(5\ \Omega\)
2. \(20\ \Omega\)
3. \(6\ A\)
4. \(3\ A\)
5. \(3\ A\)
6. \(30\ V\)
7. \(60\ V\)
8. \(30\ V\)
9. \(30\ V\)
10. \(180\ W\)