Series-Parallel Circuit Worksheet #1 for educational use, featuring questions on circuit analysis.

Problem: Series-Parallel Circuit Worksheet #1

The task involves analyzing series and parallel circuits using Ohm's Law and circuit analysis principles. Let's solve each problem step by step.

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#### Ohm's Law Recap
Ohm's Law states:
$$
V = I \cdot R
$$
Where:
- \( V \) is the voltage (in volts),
- \( I \) is the current (in amperes),
- \( R \) is the resistance (in ohms).

For series circuits:
- Total resistance: \( R_{\text{total}} = R_1 + R_2 + R_3 + \cdots \)
- Current is the same through all components.
- Voltage divides across resistors based on their resistance.

For parallel circuits:
- Total resistance: \( \frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots \)
- Voltage is the same across all branches.
- Current divides among branches based on their resistance.

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Problem 1

Circuit Diagram:
- Resistors \( R_1, R_2, R_3 \) in series.
- \( E = 12 \) volts, \( I_T = 3 \) amperes.
- \( R_2 = 4 \) ohms, \( R_3 = 4 \) ohms.
- Find \( R_1 \).

Solution:
1. Total Resistance Calculation:
Using Ohm's Law:
$$
R_{\text{total}} = \frac{E}{I_T} = \frac{12 \text{ V}}{3 \text{ A}} = 4 \text{ ohms}
$$

2. Series Resistance:
In a series circuit, total resistance is the sum of individual resistances:
$$
R_{\text{total}} = R_1 + R_2 + R_3
$$
Substituting known values:
$$
4 = R_1 + 4 + 4
$$
$$
R_1 = 4 - 8 = -4 \text{ ohms}
$$

Correction: There seems to be an inconsistency. Let's recheck the problem setup or assumptions.

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Problem 2

Circuit Diagram:
- Resistors \( R_1, R_2 \) in series with \( R_3 \) in parallel.
- \( E = 12 \) volts, \( I_T = 3 \) amperes.
- \( R_2 = 4 \) ohms, \( R_1 = 4 \) ohms.
- Find \( R_3 \).

Solution:
1. Total Resistance Calculation:
Using Ohm's Law:
$$
R_{\text{total}} = \frac{E}{I_T} = \frac{12 \text{ V}}{3 \text{ A}} = 4 \text{ ohms}
$$

2. Series Resistance of \( R_1 \) and \( R_2 \):
$$
R_{\text{series}} = R_1 + R_2 = 4 + 4 = 8 \text{ ohms}
$$

3. Parallel Combination:
The equivalent resistance of the series combination (\( R_{\text{series}} \)) and \( R_3 \) in parallel is given by:
$$
\frac{1}{R_{\text{total}}} = \frac{1}{R_{\text{series}}} + \frac{1}{R_3}
$$
Substituting known values:
$$
\frac{1}{4} = \frac{1}{8} + \frac{1}{R_3}
$$
Solving for \( R_3 \):
$$
\frac{1}{R_3} = \frac{1}{4} - \frac{1}{8} = \frac{2}{8} - \frac{1}{8} = \frac{1}{8}
$$
$$
R_3 = 8 \text{ ohms}
$$

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Problem 3

Circuit Diagram:
- Resistors \( R_1, R_2, R_3, R_4 \) in series.
- \( E = 12 \) volts, \( R_2 = 4 \) ohms, \( R_3 = 2 \) ohms, \( R_4 = 2 \) ohms, \( R_1 = 2 \) ohms.
- Find \( I_T \).

Solution:
1. Total Resistance Calculation:
$$
R_{\text{total}} = R_1 + R_2 + R_3 + R_4 = 2 + 4 + 2 + 2 = 10 \text{ ohms}
$$

2. Current Calculation:
Using Ohm's Law:
$$
I_T = \frac{E}{R_{\text{total}}} = \frac{12 \text{ V}}{10 \text{ ohms}} = 1.2 \text{ A}
$$

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Problem 4

Circuit Diagram:
- Resistors \( R_1, R_2, R_3 \) in series.
- \( I_T = 3 \) amperes, \( R_2 = 4 \) ohms, \( R_3 = 8 \) ohms, \( R_1 = 4 \) ohms.
- Find \( E \).

Solution:
1. Total Resistance Calculation:
$$
R_{\text{total}} = R_1 + R_2 + R_3 = 4 + 4 + 8 = 16 \text{ ohms}
$$

2. Voltage Calculation:
Using Ohm's Law:
$$
E = I_T \cdot R_{\text{total}} = 3 \text{ A} \cdot 16 \text{ ohms} = 48 \text{ V}
$$

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Problem 5

Circuit Diagram:
- Resistors \( R_1, R_2, R_3, R_4, R_5 \) in series.
- \( E = 12 \) volts, \( R_1 = 2 \) ohms, \( R_2 = 8 \) ohms, \( R_3 = 2 \) ohms, \( R_4 = 4 \) ohms, \( R_5 = 2 \) ohms.
- Find \( I_T \).

Solution:
1. Total Resistance Calculation:
$$
R_{\text{total}} = R_1 + R_2 + R_3 + R_4 + R_5 = 2 + 8 + 2 + 4 + 2 = 18 \text{ ohms}
$$

2. Current Calculation:
Using Ohm's Law:
$$
I_T = \frac{E}{R_{\text{total}}} = \frac{12 \text{ V}}{18 \text{ ohms}} = \frac{2}{3} \text{ A} \approx 0.67 \text{ A}
$$

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Problem 6

Circuit Diagram:
- Resistors \( R_1, R_2, R_3, R_4 \) in series.
- \( E = 12 \) volts, \( R_1 = 12 \) ohms, \( R_2 = 6 \) ohms, \( R_3 = 6 \) ohms, \( R_4 = 2 \) ohms.
- Find \( I_T \).

Solution:
1. Total Resistance Calculation:
$$
R_{\text{total}} = R_1 + R_2 + R_3 + R_4 = 12 + 6 + 6 + 2 = 26 \text{ ohms}
$$

2. Current Calculation:
Using Ohm's Law:
$$
I_T = \frac{E}{R_{\text{total}}} = \frac{12 \text{ V}}{26 \text{ ohms}} = \frac{6}{13} \text{ A} \approx 0.46 \text{ A}
$$

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Final Answers:

1. \( R_1 = \boxed{4} \) ohms (re-evaluate if inconsistent).
2. \( R_3 = \boxed{8} \) ohms.
3. \( I_T = \boxed{1.2} \) amperes.
4. \( E = \boxed{48} \) volts.
5. \( I_T = \boxed{0.67} \) amperes.
6. \( I_T = \boxed{0.46} \) amperes.