⚡ Chapter 41: Parallel Combination of Resistors (Class XII)
🔷 1. Introduction
In electrical circuits, resistors can be connected in different ways depending on the desired current and voltage distribution. One of the most common arrangements is the Parallel Combination of Resistors.
In a parallel combination, all resistors are connected across the same two terminals of the source. As a result, each resistor experiences the same potential difference (voltage), while the electric current divides among the different branches according to their resistances.
🔷 2. What is a Parallel Combination?
Ethan: Professor, what is meant by a parallel combination of resistors?
Professor: A parallel combination is an arrangement in which the ends of all resistors are connected to the same two junctions, providing multiple paths for the flow of electric current.
Academic Definition
A parallel combination of resistors is an arrangement in which all resistors are connected across the same two terminals so that each resistor has the same potential difference across it.
🔷 3. Characteristics of Parallel Combination
Ethan: Professor, what are the main characteristics of a parallel circuit?
Professor: A parallel circuit has the following characteristics.
- There are multiple paths for current.
- The potential difference across every resistor is the same.
- The total current is divided among the branches.
- The equivalent resistance is less than the smallest individual resistance.
- If one branch is disconnected, the remaining branches continue to operate.
🔷 4. Potential Difference in Parallel Combination
Ethan: Professor, does every resistor receive the same voltage?
Professor: Yes. Since all resistors are connected across the same two terminals of the source, the potential difference across each resistor is identical.
V = V₁ = V₂ = V₃ = ···
🔷 5. Current in Parallel Combination
Ethan: Professor, how is the current distributed in a parallel circuit?
Professor: The current supplied by the battery is divided among the branches. According to Kirchhoff's Current Law, the total current equals the sum of the currents in all branches.
I = I₁ + I₂ + I₃ + ···
🔷 6. Derivation of Equivalent Resistance
Ethan: Professor, how do we calculate the equivalent resistance of resistors connected in parallel?
Professor: Let three resistors R₁, R₂ and R₃ be connected in parallel.
Since the voltage across each resistor is the same,
V = I₁R₁
V = I₂R₂
V = I₃R₃
The total current is,
I = I₁ + I₂ + I₃
Using Ohm's Law,
I = V/R
Substituting the branch currents,
V/R = V/R₁ + V/R₂ + V/R₃
Dividing both sides by V,
1/R = 1/R₁ + 1/R₂ + 1/R₃
For n resistors,
1/Req = 1/R₁ + 1/R₂ + 1/R₃ + ··· + 1/Rn
🔷 7. Special Case of Two Resistors in Parallel
Ethan: Professor, is there a shortcut formula for two resistors?
Professor: Yes. For only two resistors connected in parallel, the equivalent resistance is:
R = (R₁ × R₂)/(R₁ + R₂)
🔷 8. Physical Meaning of Parallel Combination
Ethan: Professor, why does the equivalent resistance decrease in a parallel combination?
Professor: Because adding more parallel branches provides additional paths for the flow of electric current. More paths allow current to flow more easily, reducing the overall opposition offered by the circuit.
🔷 9. Example
Ethan: Professor, suppose two resistors of 6 Ω and 3 Ω are connected in parallel. What is the equivalent resistance?
Professor:
R = (6 × 3)/(6 + 3)
R = 18/9
R = 2 Ω
🔷 10. Advantages of Parallel Combination
- Each resistor receives the full supply voltage.
- If one branch fails, the remaining branches continue to function.
- The equivalent resistance decreases.
- Additional electrical devices can be connected independently.
- Current is distributed according to the resistance of each branch.
🔷 11. Disadvantages of Parallel Combination
- Requires more connecting wires.
- Circuit construction is comparatively more complex.
- Total current drawn from the source increases as more branches are added.
🔷 12. Applications
- Household electrical wiring.
- Electrical distribution systems.
- Automobile electrical circuits.
- Power distribution networks.
- Electronic appliances.
- Industrial electrical installations.
📦 13. Important Results (Must Remember)
- All resistors have the same potential difference.
- The total current is the sum of branch currents.
- Equivalent resistance is calculated using the reciprocal formula.
- Equivalent resistance is always less than the smallest individual resistance.
- Adding more resistors in parallel decreases the equivalent resistance.
- If one branch is disconnected, the remaining branches continue to operate.
- Parallel combination is widely used in household wiring.
🧠 14. Conceptual Questions
🔹 Q1
Ethan: Why is the voltage the same across all resistors connected in parallel?
Professor: Because each resistor is connected directly across the same two terminals of the source.
🔹 Q2
Ethan: How is the total current distributed in a parallel circuit?
Professor: The total current divides among the branches, and the sum of all branch currents equals the current supplied by the source.
🔹 Q3
Ethan: Why is the equivalent resistance smaller than the smallest resistor?
Professor: Because additional branches provide more paths for current, reducing the overall opposition to its flow.
🔹 Q4
Ethan: What happens if one resistor is removed from a parallel circuit?
Professor: The remaining branches continue to operate normally because each branch has an independent current path.
🔹 Q5
Ethan: Why is household wiring connected in parallel?
Professor: Because every appliance receives the full supply voltage and can operate independently of the others.
🔷 15. Comparison Between Series and Parallel Combination
| Series Combination | Parallel Combination |
|---|---|
| Only one path for current. | Multiple paths for current. |
| Current remains the same. | Voltage remains the same. |
| Voltage is divided. | Current is divided. |
| Equivalent resistance increases. | Equivalent resistance decreases. |
| Failure of one resistor stops the entire circuit. | Failure of one branch does not affect the others. |
🔷 16. Summary
In a parallel combination, all resistors are connected across the same two terminals, so each resistor experiences the same potential difference while the total current divides among the different branches. The equivalent resistance is obtained using the reciprocal formula and is always less than the smallest individual resistance. Due to its reliability and ability to provide the same voltage to every device, the parallel combination is extensively used in household wiring, power distribution systems, and modern electrical installations.
✨ End of Topic: Parallel Combination of Resistors ✨
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