Worksheet for solving parallel circuit problems using Ohm's Law and parallel circuit rules.

Let's solve each of these parallel circuit problems step by step using Ohm’s Law and the rules for parallel circuits.

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🔧 Key Rules for Parallel Circuits:

1. Voltage across each branch is the same as the total voltage:
$$
V_1 = V_2 = \dots = V_T
$$

2. Currents add up:
$$
I_T = I_1 + I_2 + \dots
$$

3. Total resistance (equivalent resistance):
$$
\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \dots
$$
or for two equal resistors:
$$
R_T = \frac{R_1 \cdot R_2}{R_1 + R_2}
$$

4. Ohm’s Law:
$$
V = I \cdot R \quad \Rightarrow \quad I = \frac{V}{R}, \quad R = \frac{V}{I}
$$

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Now let's solve each problem:

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✔ Problem 1

Circuit: Two 10Ω resistors in parallel. No voltage or current given.

We are to find: $ R_T $

$$
\frac{1}{R_T} = \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = \frac{1}{5}
\Rightarrow R_T = 5\,\Omega
$$

✔ Answer: $ R_T = 5\,\Omega $

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✔ Problem 2

Given: $ V_T = 20\,V $, $ R_1 = 20\,\Omega $, $ R_2 = 20\,\Omega $

Find: $ R_T $, $ I_T $

#### Step 1: Total Resistance
$$
\frac{1}{R_T} = \frac{1}{20} + \frac{1}{20} = \frac{2}{20} = \frac{1}{10}
\Rightarrow R_T = 10\,\Omega
$$

#### Step 2: Total Current
Use Ohm’s Law:
$$
I_T = \frac{V_T}{R_T} = \frac{20\,V}{10\,\Omega} = 2\,A
$$

✔ Answer: $ R_T = 10\,\Omega $, $ I_T = 2\,A $

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✔ Problem 3

Given: $ V_T = 40\,V $, $ R_1 = 20\,\Omega $, $ R_2 = ? $, total current $ I_T = 4\,A $

Find: $ V_1, I_1, R_T, V_2, I_2, R_2 $

#### Step 1: Voltage
In parallel: $ V_1 = V_2 = V_T = 40\,V $

✔ $ V_1 = 40\,V $, $ V_2 = 40\,V $

#### Step 2: Current through $ R_1 $
$$
I_1 = \frac{V_1}{R_1} = \frac{40\,V}{20\,\Omega} = 2\,A
$$

#### Step 3: Current through $ R_2 $
$$
I_T = I_1 + I_2 \Rightarrow I_2 = I_T - I_1 = 4\,A - 2\,A = 2\,A
$$

#### Step 4: Resistance $ R_2 $
$$
R_2 = \frac{V_2}{I_2} = \frac{40\,V}{2\,A} = 20\,\Omega
$$

#### Step 5: Total Resistance
$$
\frac{1}{R_T} = \frac{1}{20} + \frac{1}{20} = \frac{2}{20} = \frac{1}{10} \Rightarrow R_T = 10\,\Omega
$$

✔ Answers:
- $ V_1 = 40\,V $
- $ I_1 = 2\,A $
- $ R_T = 10\,\Omega $
- $ V_2 = 40\,V $
- $ I_2 = 2\,A $
- $ R_2 = 20\,\Omega $

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✔ Problem 4

Circuit: Two 30Ω resistors in parallel. Total current $ I_T = 4\,A $

Find: $ R_T $, $ V_T $

#### Step 1: Total Resistance
$$
\frac{1}{R_T} = \frac{1}{30} + \frac{1}{30} = \frac{2}{30} = \frac{1}{15}
\Rightarrow R_T = 15\,\Omega
$$

#### Step 2: Total Voltage
$$
V_T = I_T \cdot R_T = 4\,A \cdot 15\,\Omega = 60\,V
$$

✔ Answer: $ R_T = 15\,\Omega $, $ V_T = 60\,V $

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✔ Problem 5

Circuit: Two 30Ω resistors in parallel. Total current $ I_T = 2\,A $

Find: $ R_T $, $ V_T $, $ V_2 $

#### Step 1: Total Resistance
Same as above:
$$
\frac{1}{R_T} = \frac{1}{30} + \frac{1}{30} = \frac{2}{30} = \frac{1}{15}
\Rightarrow R_T = 15\,\Omega
$$

#### Step 2: Total Voltage
$$
V_T = I_T \cdot R_T = 2\,A \cdot 15\,\Omega = 30\,V
$$

#### Step 3: $ V_2 $
In parallel, voltage is same across both branches:
$$
V_2 = V_T = 30\,V
$$

✔ Answers:
- $ R_T = 15\,\Omega $
- $ V_T = 30\,V $
- $ V_2 = 30\,V $

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✔ Problem 6

Given: $ V_T = 80\,V $, $ R_1 = 40\,\Omega $, $ R_2 = 40\,\Omega $

Find: $ R_T $, $ I_T $, $ V_1 $, $ V_2 $

#### Step 1: Total Resistance
$$
\frac{1}{R_T} = \frac{1}{40} + \frac{1}{40} = \frac{2}{40} = \frac{1}{20}
\Rightarrow R_T = 20\,\Omega
$$

#### Step 2: Total Current
$$
I_T = \frac{V_T}{R_T} = \frac{80\,V}{20\,\Omega} = 4\,A
$$

#### Step 3: Voltages
In parallel: $ V_1 = V_2 = V_T = 80\,V $

✔ Answers:
- $ R_T = 20\,\Omega $
- $ I_T = 4\,A $
- $ V_1 = 80\,V $
- $ V_2 = 80\,V $

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📌 Final Answers Summary:

| Problem | $ R_T $ | $ I_T $ | $ V_T $ | $ V_1 $ | $ V_2 $ | $ I_1 $ | $ I_2 $ | $ R_2 $ |
|--------|----------|----------|----------|----------|----------|----------|----------|----------|
| 1 | 5 Ω | — | — | — | — | — | — | — |
| 2 | 10 Ω | 2 A | 20 V | — | — | — | — | — |
| 3 | 10 Ω | 4 A | 40 V | 40 V | 40 V | 2 A | 2 A | 20 Ω |
| 4 | 15 Ω | 4 A | 60 V | — | — | — | — | — |
| 5 | 15 Ω | 2 A | 30 V | — | 30 V | — | — | — |
| 6 | 20 Ω | 4 A | 80 V | 80 V | 80 V | — | — | — |

> Note: In problems where values aren't asked, they are omitted.

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Let me know if you'd like a diagram explanation or help with series-parallel combinations!