Parallel Circuit Worksheet #2 with six diagrams illustrating series-parallel circuits for educational purposes.

Let’s solve each problem one by one. These are all parallel circuits, so we’ll use the rules for parallel circuits:

- Voltage (E) is the same across all branches.
- Total current (Iₜ) = sum of currents in each branch.
- For resistors in parallel: 1/Rₜ = 1/R₁ + 1/R₂ + 1/R₃ + ...
- Ohm’s Law: I = E / R

---

Problem 1:

Given:
- E = 12 volts
- R₁ = 4 ohms
- R₂ = 12 ohms
- R₃ = 4 ohms
- R₄ = 12 ohms

We need to find Iₜ (total current).

Step 1: Find total resistance Rₜ.

Since all resistors are in parallel:

1/Rₜ = 1/4 + 1/12 + 1/4 + 1/12

Find common denominator (12):

= 3/12 + 1/12 + 3/12 + 1/12 = 8/12 = 2/3

So Rₜ = 3/2 = 1.5 ohms

Step 2: Use Ohm’s Law to find Iₜ

Iₜ = E / Rₜ = 12 / 1.5 = 8 amperes

✔ Answer for #1: 8 A

---

Problem 2:

Given:
- Iₜ = 4 amperes
- R₁ = 12 ohms
- R₂ = 12 ohms
- R₃ = 12 ohms
- R₄ = 12 ohms

All resistors equal → easy!

In parallel, if all resistors are same value, total resistance Rₜ = R / n, where n = number of resistors.

Rₜ = 12 / 4 = 3 ohms

Now use Ohm’s Law: E = Iₜ × Rₜ = 4 × 3 = 12 volts

✔ Answer for #2: 12 V

---

Problem 3:

Given:
- Iₜ = 1 ampere
- R₁ = 2 ohms
- R₂ = 4 ohms
- R₃ = 6 ohms
- R₄ = 12 ohms

Need to find E (voltage).

First, find Rₜ:

1/Rₜ = 1/2 + 1/4 + 1/6 + 1/12

Common denominator = 12:

= 6/12 + 3/12 + 2/12 + 1/12 = 12/12 = 1

So Rₜ = 1 ohm

Then E = Iₜ × Rₜ = 1 × 1 = 1 volt

✔ Answer for #3: 1 V

---

Problem 4:

Given:
- E = 12 volts
- R₁ = 8 ohms
- R₂ = 8 ohms
- R₃ = 4 ohms

Need to find Iₜ.

First, find Rₜ:

1/Rₜ = 1/8 + 1/8 + 1/4

= 1/8 + 1/8 + 2/8 = 4/8 = 1/2

So Rₜ = 2 ohms

Iₜ = E / Rₜ = 12 / 2 = 6 amperes

✔ Answer for #4: 6 A

---

Problem 5:

Given:
- E = 12 volts
- Iₜ = 4 amperes
- R₁ = 12 ohms
- R₂ = 12 ohms
- R₃ = ? (we need to find it)

This is a trickier one — we know total current and voltage, so we can find total resistance first.

Rₜ = E / Iₜ = 12 / 4 = 3 ohms

Now, since R₁ and R₂ are both 12 ohms in parallel with R₃:

1/Rₜ = 1/R₁ + 1/R₂ + 1/R₃

Plug in known values:

1/3 = 1/12 + 1/12 + 1/R₃

1/3 = 2/12 + 1/R₃ → 1/3 = 1/6 + 1/R₃

Subtract 1/6 from both sides:

1/3 - 1/6 = 1/R₃ → 2/6 - 1/6 = 1/6 = 1/R₃

So R₃ = 6 ohms

✔ Answer for #5: 6 ohms

---

Problem 6:

Given:
- Iₜ = 2 amperes
- R₁ = 24 ohms
- R₂ = 24 ohms
- R₃ = 12 ohms

Need to find E.

First, find Rₜ:

1/Rₜ = 1/24 + 1/24 + 1/12

= 1/24 + 1/24 + 2/24 = 4/24 = 1/6

So Rₜ = 6 ohms

E = Iₜ × Rₜ = 2 × 6 = 12 volts

✔ Answer for #6: 12 V

---

Final Answer:
1. 8 amperes
2. 12 volts
3. 1 volt
4. 6 amperes
5. 6 ohms
6. 12 volts