Parallel Circuit Problems Worksheet with diagrams and calculations.

Problem Analysis:

The task involves solving parallel circuit problems using the concept of equivalent resistance. The goal is to calculate the total current flowing through each circuit when a voltage source \( V \) is applied. The circuits are composed of resistors in parallel configurations, and we need to use the formula for equivalent resistance in parallel circuits:

\[
\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}
\]

Once the equivalent resistance \( R_{eq} \) is determined, the total current \( I \) can be calculated using Ohm's Law:

\[
I = \frac{V}{R_{eq}}
\]

Step-by-Step Solution:

#### Circuit (a):
- Resistors: \( R_1 = 6 \Omega \), \( R_2 = 3 \Omega \)
- Voltage: \( V = 12 \text{ V} \)

1. Calculate the equivalent resistance \( R_{eq} \):
\[
\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{6} + \frac{1}{3}
\]
\[
\frac{1}{R_{eq}} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}
\]
\[
R_{eq} = 2 \Omega
\]

2. Calculate the total current \( I \):
\[
I = \frac{V}{R_{eq}} = \frac{12}{2} = 6 \text{ A}
\]

Answer for Circuit (a):
\[
\boxed{6 \text{ A}}
\]

#### Circuit (b):
- Resistors: \( R_1 = 4 \Omega \), \( R_2 = 8 \Omega \)
- Voltage: \( V = 12 \text{ V} \)

1. Calculate the equivalent resistance \( R_{eq} \):
\[
\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{4} + \frac{1}{8}
\]
\[
\frac{1}{R_{eq}} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8}
\]
\[
R_{eq} = \frac{8}{3} \Omega
\]

2. Calculate the total current \( I \):
\[
I = \frac{V}{R_{eq}} = \frac{12}{\frac{8}{3}} = 12 \times \frac{3}{8} = \frac{36}{8} = 4.5 \text{ A}
\]

Answer for Circuit (b):
\[
\boxed{4.5 \text{ A}}
\]

#### Circuit (c):
- Resistors: \( R_1 = 10 \Omega \), \( R_2 = 5 \Omega \)
- Voltage: \( V = 12 \text{ V} \)

1. Calculate the equivalent resistance \( R_{eq} \):
\[
\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{10} + \frac{1}{5}
\]
\[
\frac{1}{R_{eq}} = \frac{1}{10} + \frac{2}{10} = \frac{3}{10}
\]
\[
R_{eq} = \frac{10}{3} \Omega
\]

2. Calculate the total current \( I \):
\[
I = \frac{V}{R_{eq}} = \frac{12}{\frac{10}{3}} = 12 \times \frac{3}{10} = \frac{36}{10} = 3.6 \text{ A}
\]

Answer for Circuit (c):
\[
\boxed{3.6 \text{ A}}
\]

#### Circuit (d):
- Resistors: \( R_1 = 2 \Omega \), \( R_2 = 6 \Omega \)
- Voltage: \( V = 12 \text{ V} \)

1. Calculate the equivalent resistance \( R_{eq} \):
\[
\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{2} + \frac{1}{6}
\]
\[
\frac{1}{R_{eq}} = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}
\]
\[
R_{eq} = \frac{3}{2} \Omega
\]

2. Calculate the total current \( I \):
\[
I = \frac{V}{R_{eq}} = \frac{12}{\frac{3}{2}} = 12 \times \frac{2}{3} = \frac{24}{3} = 8 \text{ A}
\]

Answer for Circuit (d):
\[
\boxed{8 \text{ A}}
\]

Final Answers:

\[
\boxed{6 \text{ A}, 4.5 \text{ A}, 3.6 \text{ A}, 8 \text{ A}}
\]