Parallel circuit problems worksheet with diagrams and calculations for voltage, current, and resistance in parallel circuits.

Problem Analysis:

The worksheet focuses on solving parallel circuit problems. In a parallel circuit:
1. The voltage across each branch is the same.
2. The current in the branches adds up to the total current.
3. The total resistance is calculated using the reciprocal formula:
\[
\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}
\]

Each problem in the worksheet provides a circuit diagram with specific resistances and voltages, and asks for calculations of:
- Voltage across each resistor (\( V_i \))
- Current through each resistor (\( I_i \))
- Total resistance (\( R_{\text{total}} \))

Solution Approach:

We will solve one of the circuits step-by-step as an example. Let's take Circuit 4a (top-right diagram).

#### Circuit 4a:
- Given:
- Voltage source: \( V_s = 12 \) V
- Resistors: \( R_1 = 20 \Omega \), \( R_2 = 20 \Omega \)

#### Step 1: Calculate the voltage across each resistor.
In a parallel circuit, the voltage across each resistor is the same as the source voltage:
\[
V_1 = V_2 = V_s = 12 \, \text{V}
\]

#### Step 2: Calculate the current through each resistor.
Using Ohm's Law (\( I = \frac{V}{R} \)):
\[
I_1 = \frac{V_1}{R_1} = \frac{12}{20} = 0.6 \, \text{A}
\]
\[
I_2 = \frac{V_2}{R_2} = \frac{12}{20} = 0.6 \, \text{A}
\]

#### Step 3: Calculate the total resistance.
Using the reciprocal formula for parallel resistors:
\[
\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2}
\]
\[
\frac{1}{R_{\text{total}}} = \frac{1}{20} + \frac{1}{20} = \frac{2}{20} = \frac{1}{10}
\]
\[
R_{\text{total}} = 10 \, \Omega
\]

#### Step 4: Verify the total current.
The total current can be calculated using Ohm's Law with the total resistance:
\[
I_{\text{total}} = \frac{V_s}{R_{\text{total}}} = \frac{12}{10} = 1.2 \, \text{A}
\]
This matches the sum of the individual currents:
\[
I_{\text{total}} = I_1 + I_2 = 0.6 + 0.6 = 1.2 \, \text{A}
\]

Final Answer for Circuit 4a:

\[
\boxed{
\begin{aligned}
V_1 &= 12 \, \text{V}, & V_2 &= 12 \, \text{V}, \\
I_1 &= 0.6 \, \text{A}, & I_2 &= 0.6 \, \text{A}, \\
R_{\text{total}} &= 10 \, \Omega, & I_{\text{total}} &= 1.2 \, \text{A}.
\end{aligned}
}
\]

General Method:

For any parallel circuit:
1. Use \( V_i = V_s \) for all branches.
2. Calculate \( I_i = \frac{V_s}{R_i} \) for each resistor.
3. Use the reciprocal formula to find \( R_{\text{total}} \).
4. Verify \( I_{\text{total}} = \sum I_i \).

This method can be applied to all the circuits in the worksheet. If you need solutions for other circuits, please specify!