Combination of Resistances

Connecting two or more resistors together in the same circuit is called a combination of resistances. The effective or total resistance of the circuit depends on how the resistors are connected.

The two main types of resistor combinations are—

👉 Series Combination

👉 Parallel Combination

👉 Equivalent Resistance

Suppose a combination of several resistors is replaced by a single resistor such that, for the same potential difference, the total current in the circuit remains unchanged. This single resistance is called the equivalent resistance of the original combination.

Equivalent resistance allows us to represent a complex resistor network by one effective resistance.

👉 Series Combination

When resistors are connected one after another so that the same electric current passes through every resistor, they are said to be connected in series.

If are connected in series, the equivalent resistance is—

Main Features of a Series Combination

• The same current flows through every resistor.
• The equivalent resistance is greater than any individual resistance in the combination.
• Adding more resistors in series increases the total resistance.

For Equal Resistors

If each resistor has resistance and there are resistors, then—

👉 Parallel Combination

When the two ends of several resistors are connected to the same two points of a circuit, the resistors are said to be connected in parallel.

In this arrangement, the potential difference across every resistor is the same.

If are connected in parallel, then—

Main Features of a Parallel Combination

• The potential difference across each branch is the same.
• The total current divides among the different branches.
• The equivalent resistance is smaller than the smallest individual resistance in the combination.

👉 Two Resistors Connected in Parallel

For two resistors and connected in parallel—

Therefore,

👉 n Equal Resistors Connected in Parallel

If each resistor has resistance , then—

Therefore,

Series vs Parallel

👉 Important Examples

✏️ Example 1: A wire has a total resistance of 10 Ω. It is cut into two equal parts, and the two parts are connected in parallel. Find the equivalent resistance.

When the wire is cut into two equal parts, the length of each part becomes half. Since resistance is directly proportional to length for the same material and cross-sectional area, the resistance of each part is—

Two equal resistors of 5 Ω each are connected in parallel, so—

Answer: 2.5 Ω

✏️ Example 2: Series Combination

Resistors of 2 Ω, 3 Ω, and 5 Ω are connected in series.

Answer: 10 Ω

✏️ Example 3: Two Resistors in Parallel

Resistors of 6 Ω and 3 Ω are connected in parallel.

Answer: 2 Ω

✏️ Example 4: Three Equal Resistors

Three resistors of 12 Ω each are connected in parallel.

Answer: 4 Ω

📚 Why Is a Parallel Combination Important?

In a parallel connection, every branch receives the same potential difference. This is why electrical appliances in domestic circuits are generally connected in parallel.

🧠 Quick Revision

⭐ Series:

⭐ Parallel:

⭐ n equal resistors in series:

⭐ n equal resistors in parallel:

⭐ Two resistors in parallel:

✍ Try It Yourself

🔹 Find the equivalent resistance when 4 Ω, 6 Ω, and 10 Ω are connected in series.

🔹 Find the equivalent resistance when 4 Ω and 12 Ω are connected in parallel.

🔹 Four equal resistors of 8 Ω each are connected in series. Find the total resistance.

🔹 Four equal resistors of 8 Ω each are connected in parallel. Find the total resistance.

🔹 Explain conceptually why the equivalent resistance of a parallel combination is smaller than the smallest individual resistance.