Series Circuit Problems Worksheet with diagrams and calculations for resistors in series.

Problem Analysis and Solution

The worksheet involves solving problems related to series circuits. In a series circuit:
1. The current is the same through all components.
2. The voltage drop across each resistor adds up to the total voltage supplied by the source.
3. The total resistance is the sum of the individual resistances.

We will solve each problem step by step using Ohm's Law:
\[ V = I \cdot R \]

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#### Problem 1:

- Given:
- \( V_{\text{source}} = 90 \, \text{V} \)
- \( R_1 = 10 \, \Omega \)
- \( R_2 = 20 \, \Omega \)

- To Find:
- \( R_T \), \( I_T \), \( I_1 \), \( I_2 \), \( V_1 \), \( V_2 \)

##### Step 1: Calculate Total Resistance (\( R_T \))
\[ R_T = R_1 + R_2 = 10 \, \Omega + 20 \, \Omega = 30 \, \Omega \]

##### Step 2: Calculate Total Current (\( I_T \))
Using Ohm's Law:
\[ I_T = \frac{V_{\text{source}}}{R_T} = \frac{90 \, \text{V}}{30 \, \Omega} = 3 \, \text{A} \]

##### Step 3: Calculate Voltage Drops (\( V_1 \) and \( V_2 \))
- For \( R_1 \):
\[ V_1 = I_T \cdot R_1 = 3 \, \text{A} \cdot 10 \, \Omega = 30 \, \text{V} \]
- For \( R_2 \):
\[ V_2 = I_T \cdot R_2 = 3 \, \text{A} \cdot 20 \, \Omega = 60 \, \text{V} \]

##### Step 4: Verify Currents (\( I_1 \) and \( I_2 \))
In a series circuit, the current is the same everywhere:
\[ I_1 = I_2 = I_T = 3 \, \text{A} \]

##### Final Answers for Problem 1:
\[ R_T = 30 \, \Omega, \, I_T = 3 \, \text{A}, \, I_1 = 3 \, \text{A}, \, I_2 = 3 \, \text{A}, \, V_1 = 30 \, \text{V}, \, V_2 = 60 \, \text{V} \]

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#### Problem 2:

- Given:
- \( V_{\text{source}} = 60 \, \text{V} \)
- \( R_1 = 6 \, \Omega \)
- \( R_2 = 14 \, \Omega \)
- \( R_3 = 10 \, \Omega \)

- To Find:
- \( R_T \), \( I_T \), \( I_1 \), \( I_2 \), \( I_3 \), \( V_1 \), \( V_2 \), \( V_3 \)

##### Step 1: Calculate Total Resistance (\( R_T \))
\[ R_T = R_1 + R_2 + R_3 = 6 \, \Omega + 14 \, \Omega + 10 \, \Omega = 30 \, \Omega \]

##### Step 2: Calculate Total Current (\( I_T \))
Using Ohm's Law:
\[ I_T = \frac{V_{\text{source}}}{R_T} = \frac{60 \, \text{V}}{30 \, \Omega} = 2 \, \text{A} \]

##### Step 3: Calculate Voltage Drops (\( V_1 \), \( V_2 \), and \( V_3 \))
- For \( R_1 \):
\[ V_1 = I_T \cdot R_1 = 2 \, \text{A} \cdot 6 \, \Omega = 12 \, \text{V} \]
- For \( R_2 \):
\[ V_2 = I_T \cdot R_2 = 2 \, \text{A} \cdot 14 \, \Omega = 28 \, \text{V} \]
- For \( R_3 \):
\[ V_3 = I_T \cdot R_3 = 2 \, \text{A} \cdot 10 \, \Omega = 20 \, \text{V} \]

##### Step 4: Verify Currents (\( I_1 \), \( I_2 \), and \( I_3 \))
In a series circuit, the current is the same everywhere:
\[ I_1 = I_2 = I_3 = I_T = 2 \, \text{A} \]

##### Final Answers for Problem 2:
\[ R_T = 30 \, \Omega, \, I_T = 2 \, \text{A}, \, I_1 = 2 \, \text{A}, \, I_2 = 2 \, \text{A}, \, I_3 = 2 \, \text{A}, \, V_1 = 12 \, \text{V}, \, V_2 = 28 \, \text{V}, \, V_3 = 20 \, \text{V} \]

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#### Problem 3:

- Given:
- \( V_{\text{source}} = 75 \, \text{V} \)
- \( R_1 = 10 \, \Omega \)
- \( V_2 = 25 \, \text{V} \)
- \( I_T = 5 \, \text{A} \)

- To Find:
- \( V_1 \), \( I_2 \), \( R_2 \)

##### Step 1: Calculate \( V_1 \)
Using the fact that the total voltage is the sum of the voltage drops:
\[ V_{\text{source}} = V_1 + V_2 \]
\[ 75 \, \text{V} = V_1 + 25 \, \text{V} \]
\[ V_1 = 75 \, \text{V} - 25 \, \text{V} = 50 \, \text{V} \]

##### Step 2: Calculate \( I_2 \)
In a series circuit, the current is the same everywhere:
\[ I_2 = I_T = 5 \, \text{A} \]

##### Step 3: Calculate \( R_2 \)
Using Ohm's Law for \( R_2 \):
\[ R_2 = \frac{V_2}{I_2} = \frac{25 \, \text{V}}{5 \, \text{A}} = 5 \, \Omega \]

##### Final Answers for Problem 3:
\[ V_1 = 50 \, \text{V}, \, I_2 = 5 \, \text{A}, \, R_2 = 5 \, \Omega \]

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#### Problem 4:

- Given:
- \( R_1 = 5 \, \Omega \)
- \( R_2 = 15 \, \Omega \)
- \( I_T = 5 \, \text{A} \)

- To Find:
- \( V_1 \), \( V_2 \), \( V_T \)

##### Step 1: Calculate \( V_1 \)
Using Ohm's Law for \( R_1 \):
\[ V_1 = I_T \cdot R_1 = 5 \, \text{A} \cdot 5 \, \Omega = 25 \, \text{V} \]

##### Step 2: Calculate \( V_2 \)
Using Ohm's Law for \( R_2 \):
\[ V_2 = I_T \cdot R_2 = 5 \, \text{A} \cdot 15 \, \Omega = 75 \, \text{V} \]

##### Step 3: Calculate \( V_T \)
The total voltage is the sum of the voltage drops:
\[ V_T = V_1 + V_2 = 25 \, \text{V} + 75 \, \text{V} = 100 \, \text{V} \]

##### Final Answers for Problem 4:
\[ V_1 = 25 \, \text{V}, \, V_2 = 75 \, \text{V}, \, V_T = 100 \, \text{V} \]

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#### Problem 5:

- Given:
- \( V_{\text{source}} = 20 \, \text{V} \)
- \( R_1 = 5 \, \Omega \)
- \( R_2 = 3 \, \Omega \)
- \( R_3 = 2 \, \Omega \)

- To Find:
- \( R_T \), \( I_T \), \( V_1 \), \( V_2 \), \( V_3 \)

##### Step 1: Calculate Total Resistance (\( R_T \))
\[ R_T = R_1 + R_2 + R_3 = 5 \, \Omega + 3 \, \Omega + 2 \, \Omega = 10 \, \Omega \]

##### Step 2: Calculate Total Current (\( I_T \))
Using Ohm's Law:
\[ I_T = \frac{V_{\text{source}}}{R_T} = \frac{20 \, \text{V}}{10 \, \Omega} = 2 \, \text{A} \]

##### Step 3: Calculate Voltage Drops (\( V_1 \), \( V_2 \), and \( V_3 \))
- For \( R_1 \):
\[ V_1 = I_T \cdot R_1 = 2 \, \text{A} \cdot 5 \, \Omega = 10 \, \text{V} \]
- For \( R_2 \):
\[ V_2 = I_T \cdot R_2 = 2 \, \text{A} \cdot 3 \, \Omega = 6 \, \text{V} \]
- For \( R_3 \):
\[ V_3 = I_T \cdot R_3 = 2 \, \text{A} \cdot 2 \, \Omega = 4 \, \text{V} \]

##### Final Answers for Problem 5:
\[ R_T = 10 \, \Omega, \, I_T = 2 \, \text{A}, \, V_1 = 10 \, \text{V}, \, V_2 = 6 \, \text{V}, \, V_3 = 4 \, \text{V} \]

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#### Problem 6:

- Given:
- \( V_{\text{source}} = 40 \, \text{V} \)
- \( V_3 = 10 \, \text{V} \)
- \( R_3 = 10 \, \Omega \)
- \( V_{\text{drop across } R_1 \text{ and } R_2} = 20 \, \text{V} \)

- To Find:
- \( I_3 \), \( I_1 \), \( V_1 \), \( R_1 \), \( R_2 \)

##### Step 1: Calculate \( I_3 \)
Using Ohm's Law for \( R_3 \):
\[ I_3 = \frac{V_3}{R_3} = \frac{10 \, \text{V}}{10 \, \Omega} = 1 \, \text{A} \]

##### Step 2: Calculate \( I_1 \)
In a series circuit, the current is the same everywhere:
\[ I_1 = I_3 = 1 \, \text{A} \]

##### Step 3: Calculate \( V_1 \)
The voltage drop across \( R_1 \) and \( R_2 \) is given as 20 V:
\[ V_1 = V_{\text{drop across } R_1 \text{ and } R_2} = 20 \, \text{V} \]

##### Step 4: Calculate \( R_1 \) and \( R_2 \)
Let \( R_1 + R_2 = R_{12} \). Using Ohm's Law:
\[ R_{12} = \frac{V_{\text{drop across } R_1 \text{ and } R_2}}{I_1} = \frac{20 \, \text{V}}{1 \, \text{A}} = 20 \, \Omega \]

Since the problem does not specify individual values for \( R_1 \) and \( R_2 \), we can only determine their sum:
\[ R_1 + R_2 = 20 \, \Omega \]

##### Final Answers for Problem 6:
\[ I_3 = 1 \, \text{A}, \, I_1 = 1 \, \text{A}, \, V_1 = 20 \, \text{V}, \, R_1 + R_2 = 20 \, \Omega \]

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Boxed Final Answers:

1. \( R_T = 30 \, \Omega, \, I_T = 3 \, \text{A}, \, I_1 = 3 \, \text{A}, \, I_2 = 3 \, \text{A}, \, V_1 = 30 \, \text{V}, \, V_2 = 60 \, \text{V} \)
2. \( R_T = 30 \, \Omega, \, I_T = 2 \, \text{A}, \, I_1 = 2 \, \text{A}, \, I_2 = 2 \, \text{A}, \, I_3 = 2 \, \text{A}, \, V_1 = 12 \, \text{V}, \, V_2 = 28 \, \text{V}, \, V_3 = 20 \, \text{V} \)
3. \( V_1 = 50 \, \text{V}, \, I_2 = 5 \, \text{A}, \, R_2 = 5 \, \Omega \)
4. \( V_1 = 25 \, \text{V}, \, V_2 = 75 \, \text{V}, \, V_T = 100 \, \text{V} \)
5. \( R_T = 10 \, \Omega, \, I_T = 2 \, \text{A}, \, V_1 = 10 \, \text{V}, \, V_2 = 6 \, \text{V}, \, V_3 = 4 \, \text{V} \)
6. \( I_3 = 1 \, \text{A}, \, I_1 = 1 \, \text{A}, \, V_1 = 20 \, \text{V}, \, R_1 + R_2 = 20 \, \Omega \)

\[
\boxed{
\begin{aligned}
1. & \, R_T = 30 \, \Omega, \, I_T = 3 \, \text{A}, \, I_1 = 3 \, \text{A}, \, I_2 = 3 \, \text{A}, \, V_1 = 30 \, \text{V}, \, V_2 = 60 \, \text{V} \\
2. & \, R_T = 30 \, \Omega, \, I_T = 2 \, \text{A}, \, I_1 = 2 \, \text{A}, \, I_2 = 2 \, \text{A}, \, I_3 = 2 \, \text{A}, \, V_1 = 12 \, \text{V}, \, V_2 = 28 \, \text{V}, \, V_3 = 20 \, \text{V} \\
3. & \, V_1 = 50 \, \text{V}, \, I_2 = 5 \, \text{A}, \, R_2 = 5 \, \Omega \\
4. & \, V_1 = 25 \, \text{V}, \, V_2 = 75 \, \text{V}, \, V_T = 100 \, \text{V} \\
5. & \, R_T = 10 \, \Omega, \, I_T = 2 \, \text{A}, \, V_1 = 10 \, \text{V}, \, V_2 = 6 \, \text{V}, \, V_3 = 4 \, \text{V} \\
6. & \, I_3 = 1 \, \text{A}, \, I_1 = 1 \, \text{A}, \, V_1 = 20 \, \text{V}, \, R_1 + R_2 = 20 \, \Omega
\end{aligned}
}
\]