SOLUTION: 1 series circuits module - Studypool - Free Printable

Let's solve each problem on the Series Circuit Worksheet #2 using Ohm’s Law and principles of series circuits.

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🔧 Key Concepts for Series Circuits:

1. Total Resistance (Rₜ):
$$
R_T = R_1 + R_2 + R_3 + \dots
$$

2. Current (Iₜ):
In a series circuit, current is the same everywhere:
$$
I_T = \frac{E}{R_T}
$$

3. Voltage (E):
$$
E = I_T \times R_T
$$

4. Ohm’s Law:
$$
V = I \times R
$$

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We'll go through each problem one by one.

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🔹 Problem 1:

Given:
- $ R_1 = 1\ \Omega $
- $ R_2 = 1\ \Omega $
- $ R_3 = 1\ \Omega $
- $ I_T = 8\ \text{A} $
- Find: $ E = ? $

#### Step 1: Total Resistance
$$
R_T = R_1 + R_2 + R_3 = 1 + 1 + 1 = 3\ \Omega
$$

#### Step 2: Use Ohm’s Law to find Voltage
$$
E = I_T \times R_T = 8\ \text{A} \times 3\ \Omega = 24\ \text{volts}
$$

✔ Answer: $ E = 24 $ volts

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🔹 Problem 2:

Given:
- $ E = 24\ \text{V} $
- $ R_1 = 2\ \Omega $
- $ R_2 = 2\ \Omega $
- $ R_3 = 2\ \Omega $
- Find: $ I_T = ? $

#### Step 1: Total Resistance
$$
R_T = 2 + 2 + 2 = 6\ \Omega
$$

#### Step 2: Current
$$
I_T = \frac{E}{R_T} = \frac{24\ \text{V}}{6\ \Omega} = 4\ \text{A}
$$

✔ Answer: $ I_T = 4 $ amperes

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🔹 Problem 3:

Given:
- $ E = 24\ \text{V} $
- $ I_T = 4\ \text{A} $
- $ R_1 = 2\ \Omega $
- $ R_2 = 1\ \Omega $
- $ R_3 = ? $

Find: $ R_3 = ? $

#### Step 1: Total Resistance
Use Ohm’s Law:
$$
R_T = \frac{E}{I_T} = \frac{24\ \text{V}}{4\ \text{A}} = 6\ \Omega
$$

#### Step 2: Solve for $ R_3 $
$$
R_T = R_1 + R_2 + R_3 \\
6 = 2 + 1 + R_3 \\
R_3 = 6 - 3 = 3\ \Omega
$$

✔ Answer: $ R_3 = 3\ \Omega $

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🔹 Problem 4:

Given:
- $ E = 12\ \text{V} $
- $ I_T = 2\ \text{A} $
- $ R_1 = 1\ \Omega $
- $ R_2 = 1\ \Omega $
- $ R_3 = 2\ \Omega $
- $ R_4 = 1\ \Omega $
- Find: $ R_2 = ? $ → Wait! But $ R_2 $ is already given as 1 ohm.

Wait — looks like it says “R2 = ___” but below it says “R2 = 1 ohm”. This may be a typo in the image.

But let's check what’s being asked.

Actually, looking at the diagram:
- It says: $ R_1 = 1\ \Omega $, $ R_2 = ? $, $ R_3 = 2\ \Omega $, $ R_4 = 1\ \Omega $
- But then it says “R2 = 1 ohm” under the diagram — that seems inconsistent.

Wait — perhaps the question is asking for $ R_2 $? But it says "R2 = ___" and lists "R2 = 1 ohm" below. That’s confusing.

Let me re-read carefully:

> E = 12 volts
> I_T = 2 amperes
> R1 = 1 ohm
> R2 = ?
> R3 = 2 ohms
> R4 = 1 ohm

Ah! So the value of $ R_2 $ is missing, and we’re to find it.

So:

#### Step 1: Total Resistance
$$
R_T = \frac{E}{I_T} = \frac{12\ \text{V}}{2\ \text{A}} = 6\ \Omega
$$

#### Step 2: Sum known resistances
$$
R_1 + R_3 + R_4 = 1 + 2 + 1 = 4\ \Omega
$$

Then:
$$
R_2 = R_T - (R_1 + R_3 + R_4) = 6 - 4 = 2\ \Omega
$$

✔ Answer: $ R_2 = 2\ \Omega $

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🔹 Problem 5:

Given:
- $ E = 24\ \text{V} $
- $ I_T = 2\ \text{A} $
- $ R_1 = 2\ \Omega $
- $ R_2 = 4\ \Omega $
- $ R_3 = 1\ \Omega $
- $ R_4 = ? $

Find: $ R_4 = ? $

#### Step 1: Total Resistance
$$
R_T = \frac{E}{I_T} = \frac{24}{2} = 12\ \Omega
$$

#### Step 2: Sum known resistances
$$
R_1 + R_2 + R_3 = 2 + 4 + 1 = 7\ \Omega
$$

Then:
$$
R_4 = 12 - 7 = 5\ \Omega
$$

✔ Answer: $ R_4 = 5\ \Omega $

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🔹 Problem 6:

Given:
- $ I_T = 3\ \text{A} $
- $ R_1 = 2\ \Omega $
- $ R_2 = 1\ \Omega $
- $ R_3 = 2\ \Omega $
- $ R_4 = 1\ \Omega $
- Find: $ E = ? $

#### Step 1: Total Resistance
$$
R_T = 2 + 1 + 2 + 1 = 6\ \Omega
$$

#### Step 2: Voltage
$$
E = I_T \times R_T = 3\ \text{A} \times 6\ \Omega = 18\ \text{V}
$$

✔ Answer: $ E = 18 $ volts

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ E = 24 $ volts |
| 2 | $ I_T = 4 $ amperes |
| 3 | $ R_3 = 3 $ ohms |
| 4 | $ R_2 = 2 $ ohms |
| 5 | $ R_4 = 5 $ ohms |
| 6 | $ E = 18 $ volts |

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📌 Explanation Recap:

All circuits are series, so:
- Total resistance is sum of all resistors.
- Current is the same throughout.
- Voltage is total voltage across the source.
- Use Ohm’s Law: $ E = I \times R $, rearranged as needed.

These problems test your ability to apply Ohm’s Law and understand how components behave in series circuits.

Let me know if you'd like this formatted as a filled-in worksheet!