Solved PARALLEL CIRCUIT PROBLEMS Use Ohms Law and the rules ... - Free Printable

To solve the problems involving parallel circuits, we will use Ohm's Law and the rules for parallel circuits. Let's go through each problem step by step.

Problem 1:

Circuit Diagram:
- Two resistors \( R_1 = 10 \Omega \) and \( R_2 = 10 \Omega \) in parallel.
- Find \( R_T \).

Solution:
For resistors in parallel:
\[
\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}
\]
Substitute the values:
\[
\frac{1}{R_T} = \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = \frac{1}{5}
\]
Thus,
\[
R_T = 5 \Omega
\]

Answer:
\[
R_T = 5 \Omega
\]

---

Problem 2:

Circuit Diagram:
- Two resistors \( R_1 = 20 \Omega \) and \( R_2 = 20 \Omega \) in parallel.
- Voltage \( V_T = 20 \text{ V} \).
- Find \( R_T \) and \( I_T \).

Solution:
First, find \( R_T \):
\[
\frac{1}{R_T} = \frac{1}{20} + \frac{1}{20} = \frac{2}{20} = \frac{1}{10}
\]
Thus,
\[
R_T = 10 \Omega
\]

Next, find \( I_T \) using Ohm's Law:
\[
I_T = \frac{V_T}{R_T} = \frac{20}{10} = 2 \text{ A}
\]

Answer:
\[
R_T = 10 \Omega, \quad I_T = 2 \text{ A}
\]

---

Problem 3:

Circuit Diagram:
- Two resistors \( R_1 = 20 \Omega \) and \( R_2 \) in parallel.
- Total voltage \( V_T = 4 \text{ V} \).
- Total current \( I_T = 4 \text{ A} \).
- Find \( V_1 \), \( I_1 \), \( I_2 \), \( R_T \), and \( R_2 \).

Solution:
Since the resistors are in parallel, the voltage across each resistor is the same:
\[
V_1 = V_2 = V_T = 4 \text{ V}
\]

Using Ohm's Law to find \( I_1 \):
\[
I_1 = \frac{V_1}{R_1} = \frac{4}{20} = 0.2 \text{ A}
\]

The total current is the sum of the currents through each branch:
\[
I_T = I_1 + I_2
\]
\[
4 = 0.2 + I_2 \implies I_2 = 3.8 \text{ A}
\]

Now, find \( R_2 \) using Ohm's Law:
\[
R_2 = \frac{V_2}{I_2} = \frac{4}{3.8} \approx 1.05 \Omega
\]

Finally, find \( R_T \):
\[
\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{20} + \frac{1}{1.05} \approx 0.05 + 0.952 \approx 1.002
\]
\[
R_T \approx \frac{1}{1.002} \approx 0.998 \Omega \approx 1 \Omega
\]

Answer:
\[
V_1 = 4 \text{ V}, \quad I_1 = 0.2 \text{ A}, \quad I_2 = 3.8 \text{ A}, \quad R_T \approx 1 \Omega, \quad R_2 \approx 1.05 \Omega
\]

---

Problem 4:

Circuit Diagram:
- Two resistors \( R_1 = 30 \Omega \) and \( R_2 = 30 \Omega \) in parallel.
- Total current \( I_T = 4 \text{ A} \).
- Find \( R_T \) and \( V_T \).

Solution:
First, find \( R_T \):
\[
\frac{1}{R_T} = \frac{1}{30} + \frac{1}{30} = \frac{2}{30} = \frac{1}{15}
\]
Thus,
\[
R_T = 15 \Omega
\]

Next, find \( V_T \) using Ohm's Law:
\[
V_T = I_T \cdot R_T = 4 \cdot 15 = 60 \text{ V}
\]

Answer:
\[
R_T = 15 \Omega, \quad V_T = 60 \text{ V}
\]

---

Problem 5:

Circuit Diagram:
- Two resistors \( R_1 = 30 \Omega \) and \( R_2 = 30 \Omega \) in parallel.
- Total current \( I_T = 2 \text{ A} \).
- Find \( R_T \) and \( V_T \).

Solution:
First, find \( R_T \):
\[
\frac{1}{R_T} = \frac{1}{30} + \frac{1}{30} = \frac{2}{30} = \frac{1}{15}
\]
Thus,
\[
R_T = 15 \Omega
\]

Next, find \( V_T \) using Ohm's Law:
\[
V_T = I_T \cdot R_T = 2 \cdot 15 = 30 \text{ V}
\]

Answer:
\[
R_T = 15 \Omega, \quad V_T = 30 \text{ V}
\]

---

Problem 6:

Circuit Diagram:
- Two resistors \( R_1 = 40 \Omega \) and \( R_2 = 40 \Omega \) in parallel.
- Total voltage \( V_T = 80 \text{ V} \).
- Find \( R_T \), \( I_T \), \( V_1 \), and \( V_2 \).

Solution:
First, find \( R_T \):
\[
\frac{1}{R_T} = \frac{1}{40} + \frac{1}{40} = \frac{2}{40} = \frac{1}{20}
\]
Thus,
\[
R_T = 20 \Omega
\]

Next, find \( I_T \) using Ohm's Law:
\[
I_T = \frac{V_T}{R_T} = \frac{80}{20} = 4 \text{ A}
\]

Since the resistors are in parallel, the voltage across each resistor is the same:
\[
V_1 = V_2 = V_T = 80 \text{ V}
\]

Answer:
\[
R_T = 20 \Omega, \quad I_T = 4 \text{ A}, \quad V_1 = 80 \text{ V}, \quad V_2 = 80 \text{ V}
\]

---

Final Answers:

1. \( R_T = 5 \Omega \)
2. \( R_T = 10 \Omega, \quad I_T = 2 \text{ A} \)
3. \( V_1 = 4 \text{ V}, \quad I_1 = 0.2 \text{ A}, \quad I_2 = 3.8 \text{ A}, \quad R_T \approx 1 \Omega, \quad R_2 \approx 1.05 \Omega \)
4. \( R_T = 15 \Omega, \quad V_T = 60 \text{ V} \)
5. \( R_T = 15 \Omega, \quad V_T = 30 \text{ V} \)
6. \( R_T = 20 \Omega, \quad I_T = 4 \text{ A}, \quad V_1 = 80 \text{ V}, \quad V_2 = 80 \text{ V} \)

\boxed{
\begin{aligned}
1. & \quad R_T = 5 \Omega \\
2. & \quad R_T = 10 \Omega, \quad I_T = 2 \text{ A} \\
3. & \quad V_1 = 4 \text{ V}, \quad I_1 = 0.2 \text{ A}, \quad I_2 = 3.8 \text{ A}, \quad R_T \approx 1 \Omega, \quad R_2 \approx 1.05 \Omega \\
4. & \quad R_T = 15 \Omega, \quad V_T = 60 \text{ V} \\
5. & \quad R_T = 15 \Omega, \quad V_T = 30 \text{ V} \\
6. & \quad R_T = 20 \Omega, \quad I_T = 4 \text{ A}, \quad V_1 = 80 \text{ V}, \quad V_2 = 80 \text{ V}
\end{aligned}
}