Resistors in Parallel: Formula and Derivation | Class 10

Quick answer: When resistors are connected in parallel, the potential difference across each is the same, and the total current is the sum of the branch currents. The equivalent resistance is given by $\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$; for two resistors, $R_p = \frac{R_1R_2}{R_1 + R_2}$. $R_p$ is always less than the smallest resistance. For example, $6\,\Omega$ and $12\,\Omega$ in parallel give $R_p = 4\,\Omega$.

When you switch on a fan at home, the lights do not go dim or switch off. Each appliance works independently and gets the full mains voltage. This is because household appliances are connected in parallel.

In CBSE Class 10 Science, Chapter 11 (Electricity), deriving the formula for resistors in parallel and why domestic circuits use parallel connections are frequently asked board exam questions.

What You Will Learn

• The two rules for resistors in parallel
• The derivation of $\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$
• The shortcut for two resistors, $R_p = \frac{R_1R_2}{R_1 + R_2}$
• How current divides between two parallel branches
• Why household circuits are connected in parallel
• A solved numerical problem

1. Rules for Resistors in Parallel

In a parallel combination, the resistors are connected between the same two points.

2. Derivation of the Formula

Step 1: Apply Ohm's law to each resistor. The potential difference across each is $V$, so:

$I_1 = \frac{V}{R_1} \qquad I_2 = \frac{V}{R_2} \qquad I_3 = \frac{V}{R_3}$

Step 2: Add the branch currents.

$I = I_1 + I_2 + I_3 = V\left(\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\right) \qquad \text{...(1)}$

Step 3: Use the equivalent resistance. If the whole combination is replaced by one resistor $R_p$ that draws the same current $I$:

$I = \frac{V}{R_p} \qquad \text{...(2)}$

Step 4: Compare (1) and (2) and divide by $V$:

$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

Result: The reciprocal of the equivalent resistance of resistors in parallel equals the sum of the reciprocals of the individual resistances.

3. Shortcut for Two Resistors

For two resistors in parallel:

$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{R_1 + R_2}{R_1R_2} \implies R_p = \frac{R_1R_2}{R_1 + R_2}$

This is often remembered as product over sum.

Key point: The equivalent resistance of a parallel combination is always less than the smallest resistance in it. For example, $1000\,\Omega$ and $1\,\Omega$ in parallel give $R_p = \frac{1000}{1001} \approx 0.999\,\Omega$.

4. How Current Divides Between Two Branches

Since $V = I_1R_1 = I_2R_2$, the branch currents are:

$I_1 = I \times \frac{R_2}{R_1 + R_2} \qquad I_2 = I \times \frac{R_1}{R_1 + R_2}$

Remember: More current flows through the smaller resistance. The current in each branch is inversely proportional to its resistance.

5. Why Household Circuits Are Connected in Parallel

Common board question: Give reasons why appliances in a house are connected in parallel and not in series.

In a series circuit, by contrast, the voltage is shared between appliances, and if one appliance stops working, the whole circuit is broken.

6. Solved Example

Question

Two resistors of $6\,\Omega$ and $12\,\Omega$ are connected in parallel to a $12\text{ V}$ battery. Find (a) the equivalent resistance, (b) the total current and (c) the current through each resistor.

Solution

Step 1: Equivalent resistance.

$R_p = \frac{6 \times 12}{6 + 12} = \frac{72}{18} = 4\,\Omega$

Step 2: Total current.

$I = \frac{V}{R_p} = \frac{12}{4} = 3\text{ A}$

Step 3: Current through each resistor. The potential difference across each is $12\text{ V}$:

$I_1 = \frac{12}{6} = 2\text{ A} \qquad I_2 = \frac{12}{12} = 1\text{ A}$

Answer: (a) $4\,\Omega$ (b) $3\text{ A}$ (c) $2\text{ A}$ through the $6\,\Omega$ resistor and $1\text{ A}$ through the $12\,\Omega$ resistor.

Check: $2 + 1 = 3\text{ A}$, which equals the total current. ✓ The smaller resistor carries more current, as expected.

7. Summary Table

8. Exam Tips and Common Mistakes

Exam Tip: In the derivation, clearly state that the potential difference is the same across each resistor in parallel.

Exam Tip: Check your answer: $R_p$ must be smaller than the smallest resistor, and the branch currents must add up to the total current.

Common Mistake: Forgetting to take the reciprocal at the end. If $\frac{1}{R_p} = \frac{1}{4}$, then $R_p = 4\,\Omega$, not $\frac{1}{4}\,\Omega$.

Common Mistake: Mixing up the rules for series and parallel. In series, the current is the same; in parallel, the voltage is the same.

Frequently Asked Questions

What is the formula for resistors in parallel?

1/Rp = 1/R₁ + 1/R₂ + 1/R₃. For two resistors, Rp = R₁R₂/(R₁ + R₂).

Is the equivalent resistance in parallel more or less than each resistor?

Less. It is always smaller than the smallest resistance in the combination.

Why are household appliances connected in parallel?

So that each gets the full supply voltage, can be switched on or off independently, and keeps working if another appliance fails.

What is the equivalent resistance of 6 Ω and 12 Ω in parallel?

(6 × 12)/(6 + 12) = 72/18 = 4 Ω.

What remains the same in a parallel circuit?

The potential difference across each resistor.