Look around any room and count the electrical devices: lights, fans, TV, phone charger, refrigerator. Every one of these operates on a circuit, and in every circuit, resistors are connected in specific arrangements — either in series (one after another, like beads on a string) or in parallel (side by side, like lanes on a highway). Understanding how these two arrangements behave differently is the foundation of all electrical engineering and one of the most important topics in CBSE Class 10 Physics, Chapter 12: Electricity.
The key question in any resistor combination is: What single resistance value (the equivalent resistance) would draw the same current from the battery as the entire combination? For resistors in series, equivalent resistance is the sum of all individual resistances. For resistors in parallel, the equivalent resistance is always less than the smallest individual resistance — adding more parallel paths makes it easier for current to flow.
This guide covers both combinations completely: definitions and properties, circuit diagrams, full derivations of equivalent resistance formulas, step-by-step experiment procedure, observation tables, a detailed comparison table, uses in real circuits, and five CBSE exam-ready FAQs. All apparatus described is manufactured and supplied by AJKANT Overseas from Ambala, India.
- 1. Resistors in Series — Definition, Diagram, Properties
- 2. Derivation of Equivalent Resistance in Series
- 3. Resistors in Parallel — Definition, Diagram, Properties
- 4. Derivation of Equivalent Resistance in Parallel
- 5. Series vs Parallel — Key Differences
- 6. Experiment Procedure — Verifying the Formulas
- 7. Observation Tables
- 8. Uses of Series and Parallel Circuits
- 9. Frequently Asked Questions (FAQ)
1. Resistors in Series — Definition, Circuit Diagram, Properties
When resistors are connected in series, they are joined end-to-end so that the same current flows through each resistor one after another. There is only one path for the current to flow, so the current has no choice but to pass through each resistor in sequence.
📈 Circuit Diagram: Resistors in Series
Key properties of series connection:
- The same current I flows through each resistor (only one path available).
- The total voltage V is shared: V = V₁ + V₂ + V₃ (Kirchhoff’s voltage law).
- The equivalent resistance Rᵉ is the sum: Rᵉ = R₁ + R₂ + R₃.
- Rᵉ is always greater than any individual resistance.
- If one resistor fails (open circuit), the entire circuit breaks — no current flows.
2. Derivation of Equivalent Resistance in Series
Let: I = current flowing through the series circuit (same through all resistors).
Step 1: By Ohm’s law for each resistor:
V₁ = I R₁ | V₂ = I R₂ | V₃ = I R₃
Step 2: By Kirchhoff’s Voltage Law (KVL) — sum of voltage drops = total EMF:
V = V₁ + V₂ + V₃
Step 3: Substituting:
V = IR₁ + IR₂ + IR₃ = I(R₁ + R₂ + R₃)
Step 4: By Ohm’s law for equivalent resistance: V = IRᵉ
IRᵉ = I(R₁ + R₂ + R₃)
Rᵉ = 4 + 6 + 10 = 20 Ω
Current I = V/Rᵉ = 20/20 = 1 A
V₁ = 1×4 = 4 V | V₂ = 1×6 = 6 V | V₃ = 1×10 = 10 V | Total = 4+6+10 = 20 V ✓
3. Resistors in Parallel — Definition, Circuit Diagram, Properties
When resistors are connected in parallel, both terminals of each resistor are connected to the same two points in the circuit. The same voltage appears across every resistor, but the current from the battery splits among the different resistors — more current flows through lower resistance paths.
📈 Circuit Diagram: Resistors in Parallel
Key properties of parallel connection:
- The same voltage V appears across each resistor (all share the same two nodes).
- The total current I is shared: I = I₁ + I₂ + I₃ (Kirchhoff’s current law).
- The equivalent resistance Rᵉ satisfies: 1/Rᵉ = 1/R₁ + 1/R₂ + 1/R₃.
- Rᵉ is always less than the smallest individual resistance.
- If one resistor fails, the other resistors continue to work normally.
4. Derivation of Equivalent Resistance in Parallel
Let: I₁, I₂, I₃ = currents through R₁, R₂, R₃ respectively. I = total current from battery.
Step 1: By Ohm’s law for each resistor (same voltage V across all):
I₁ = V/R₁ | I₂ = V/R₂ | I₃ = V/R₃
Step 2: By Kirchhoff’s Current Law (KCL) — total current = sum of branch currents:
I = I₁ + I₂ + I₃
Step 3: Substituting:
I = V/R₁ + V/R₂ + V/R₃ = V(1/R₁ + 1/R₂ + 1/R₃)
Step 4: By Ohm’s law for equivalent resistance: I = V/Rᵉ, so V/Rᵉ = V(1/R₁ + 1/R₂ + 1/R₃)
Worked Example: R₁ = 6 Ω, R₂ = 12 Ω, R₃ = 4 Ω connected in parallel across 12 V battery.
1/Rᵉ = 1/6 + 1/12 + 1/4 = 2/12 + 1/12 + 3/12 = 6/12 = 1/2
Rᵉ = 2 Ω (less than smallest individual R = 4 Ω ✓)
Total I = V/Rᵉ = 12/2 = 6 A
I₁ = 12/6 = 2 A | I₂ = 12/12 = 1 A | I₃ = 12/4 = 3 A | Total = 2+1+3 = 6 A ✓
5. Series vs Parallel Resistors — Key Differences
| Property | Series Connection | Parallel Connection |
|---|---|---|
| Current | Same through all resistors (I₁ = I₂ = I₃ = I) | Different through each; sums to total (I = I₁ + I₂ + I₃) |
| Voltage | Splits across resistors (V = V₁ + V₂ + V₃) | Same across all resistors (V₁ = V₂ = V₃ = V) |
| Equivalent Resistance | Rᵉ = R₁ + R₂ + R₃ (sum) | 1/Rᵉ = 1/R₁ + 1/R₂ + 1/R₃ (reciprocal sum) |
| Rᵉ compared to individual R | Rᵉ is always greater than any R | Rᵉ is always less than any R |
| Adding more resistors | Rᵉ increases; current decreases | Rᵉ decreases; total current increases |
| If one resistor fails (open) | Entire circuit fails; no current flows | Other resistors continue to work normally |
| If one resistor fails (short) | Remaining resistors carry more voltage | Total resistance drops; dangerously high current |
| Brightness of bulbs | Equal and dimmer (each gets a fraction of V) | Equal and brighter (each gets full V) |
| Examples in daily life | Old-style fairy lights, fuses in series, voltage dividers | Household electrical wiring, car electrical systems, appliances on mains |
6. Experiment Procedure — Verifying the Equivalent Resistance Formulas
Aim: To verify the laws of combination of resistors (series and parallel) using a battery, ammeter, voltmeter, and known resistors.
Apparatus Required
- Three known resistors (e.g., R₁ = 4 Ω, R₂ = 6 Ω, R₃ = 10 Ω) — labelled resistance coils or colour-coded resistors
- Battery (6V) or battery eliminator with plug key K
- Ammeter (0–3 A range) connected in series in the circuit
- Voltmeter (0–10 V range) connected in parallel across the combination
- Connecting wires, rheostat (for adjusting current)
Part A: Series Combination
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Connect R₁ AloneConnect R₁ alone in the circuit (with ammeter in series, voltmeter across R₁, battery, and key K). Close K. Record ammeter reading (I) and voltmeter reading (V₁). Calculate R₁ = V₁/I. Record.
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Connect R₁ and R₂ in SeriesConnect R₁ and R₂ in series (end-to-end). Connect ammeter in series, voltmeter across both. Close K. Record I and V. Calculate Rᵉ(observed) = V/I. Compare with Rᵉ(theoretical) = R₁ + R₂. Record.
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Connect R₁, R₂, and R₃ in SeriesConnect all three in series. Record I and V. Calculate Rᵉ(observed) = V/I. Compare with Rᵉ(theoretical) = R₁ + R₂ + R₃. The two values should match (within experimental error), verifying the series law.
Part B: Parallel Combination
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Connect R₁ and R₂ in ParallelConnect R₁ and R₂ in parallel (both terminals of each connected to the same two points). Connect the ammeter in series with the parallel combination, voltmeter across the combination. Close K. Record total current I and voltage V. Calculate Rᵉ(observed) = V/I. Compare with Rᵉ(theoretical) = R₁R₂/(R₁+R₂).
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Connect R₁, R₂, and R₃ in ParallelConnect all three in parallel. Record I and V. Calculate Rᵉ(observed) = V/I. Compare with Rᵉ(theoretical) from 1/Rᵉ = 1/R₁ + 1/R₂ + 1/R₃. The two values should match, verifying the parallel law.
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Measure Individual Branch Currents (Optional)For the parallel circuit, temporarily move the ammeter into each individual branch (in series with R₁, then R₂, then R₃) to measure I₁, I₂, I₃ separately. Verify that I₁ + I₂ + I₃ = I (total) as predicted by KCL.
7. Observation Tables
Battery EMF: _____ V | R₁ = _____ Ω | R₂ = _____ Ω | R₃ = _____ Ω
Table 1 — Series Combination
| Connection | Voltmeter V (Volts) | Ammeter I (Amperes) | Rᵉ Observed = V/I (Ω) | Rᵉ Theoretical (Ω) | Result |
|---|---|---|---|---|---|
| R₁ alone | ____ | ____ | ____ | ____ | ✓ |
| R₁ + R₂ series | ____ | ____ | ____ | R₁+R₂=____ | ✓ |
| R₁ + R₂ + R₃ series | ____ | ____ | ____ | R₁+R₂+R₃=____ | ✓ |
Table 2 — Parallel Combination
| Connection | Voltmeter V (Volts) | Ammeter I (Amperes) | Rᵉ Observed = V/I (Ω) | Rᵉ Theoretical (Ω) | Result |
|---|---|---|---|---|---|
| R₁ || R₂ parallel | ____ | ____ | ____ | R₁R₂/(R₁+R₂)=____ | ✓ |
| R₁ || R₂ || R₃ parallel | ____ | ____ | ____ | 1/Rᵉ=1/R₁+1/R₂+1/R₃ | ✓ |
Parallel: Rᵉ(observed) ≈ Rᵉ(theoretical) from 1/Rᵉ = 1/R₁ + 1/R₂ + 1/R₃ ✓
8. Uses of Series and Parallel Circuits in Daily Life
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9. Frequently Asked Questions (FAQ)
Series: Rᵉ = R₁ + R₂ + R₃ + … (simply add all resistances). For n equal resistors of resistance R each: Rᵉ = nR.
Parallel: 1/Rᵉ = 1/R₁ + 1/R₂ + 1/R₃ + … (add reciprocals). For two resistors: Rᵉ = R₁R₂/(R₁+R₂) (product over sum). For n equal resistors of resistance R each: Rᵉ = R/n.
Key point: In series, Rᵉ is always greater than any individual resistance. In parallel, Rᵉ is always less than the smallest individual resistance.
When resistors are connected in parallel, each additional resistor provides an additional path for current to flow. More current paths mean less total opposition to current flow, i.e., less total resistance. Mathematically, from 1/Rᵉ = 1/R₁ + 1/R₂, since both 1/R₁ and 1/R₂ are positive, 1/Rᵉ > 1/R₁ and 1/Rᵉ > 1/R₂. Taking reciprocals reverses the inequality: Rᵉ < R₁ and Rᵉ < R₂. So Rᵉ is always smaller than either individual resistance. Physically: even if R₂ is a very large resistance, adding it in parallel provides at least some extra path for current, which always reduces the total resistance below R₁.
Household appliances are connected in parallel for three key reasons: (1) Each appliance gets the full supply voltage (230V): In series, the voltage would be divided among all appliances, so none would work at its rated voltage. (2) Independent operation: In parallel, each appliance can be switched on or off independently without affecting others. In series, turning off one appliance breaks the entire circuit. (3) Failure isolation: If one appliance fails in parallel, others continue to work. In series, one failure stops all appliances.
Let each resistor have resistance R.
Series: Rᵉ(series) = R + R = 2R
Parallel: Rᵉ(parallel) = R×R/(R+R) = R²/2R = R/2
Ratio: Rᵉ(series) / Rᵉ(parallel) = 2R / (R/2) = 2R × 2/R = 4
So the series equivalent resistance is 4 times the parallel equivalent resistance. This is a common CBSE numerical question: “The ratio of series to parallel equivalent resistance for two equal resistors is 4:1.”
Adding more resistors in series: The total resistance Rᵉ increases (Rᵉ = R₁ + R₂ + …), so the total current I = V/Rᵉ decreases. Each existing resistor carries the same current as before, but that current is now smaller.
Adding more resistors in parallel: The total resistance Rᵉ decreases (1/Rᵉ = 1/R₁ + 1/R₂ + …), so the total current I = V/Rᵉ increases. Each existing resistor still carries the same current as before (same voltage across each), but the battery must now supply more total current. This is why adding more appliances to a household circuit increases the total current drawn and can trip the circuit breaker.
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