Three resistors of resistances 11 Ω, 22 Ω and 33 Ω are connected in parallel. Their equivalent resistance is equal to
- (a)66 Ω
- (b)22 Ω
- (c)12 Ω
- (d)6 Ω
Correct — D, 6 Ω. For resistors in parallel the reciprocals add: 1/R = 1/R1 + 1/R2 + 1/R3. Put the three values over a common denominator of 66. One eleventh is 6/66, one twenty-second is 3/66, one thirty-third is 2/66, and the three add to 11/66, which is 1/6. Invert and the equivalent resistance is 6 Ω. There is a faster route that needs no arithmetic at all. Adding a resistor in parallel always opens another path for current, so the combination must be easier to push current through than any single branch — the equivalent resistance of a parallel set is always smaller than the smallest resistor in it. The smallest here is 11 Ω, so anything at or above 11 is impossible on sight. That kills 66 Ω, 22 Ω and 12 Ω in one stroke and leaves 6 Ω standing, before the fractions are touched. Keep the contrast with series in mind: in series the resistances simply add, and 11 plus 22 plus 33 would give 66 Ω, which is the option planted for the candidate who reaches for the wrong rule.
- (a)66 Ω — The series answer. Adding 11, 22 and 33 gives 66, which is what the combination would be if the resistors were joined end to end instead of side by side.
- (b)22 Ω — Simply one of the given values, and larger than the smallest resistor in the set. No parallel combination can exceed its smallest branch.
- (c)12 Ω — The most tempting wrong figure because it looks like a reduced value, but it is still above 11 Ω. It appears to come from averaging or from a slip in taking the reciprocal.
In a series combination the same current passes through every resistor and the potential differences add, so the equivalent resistance is the sum. In a parallel combination every resistor has the same potential difference across it and the currents add, so it is the reciprocals of the resistances that add: 1/Rp = 1/R1 + 1/R2 + 1/R3. NCERT states the rule as the reciprocal of the equivalent resistance of a group of resistances joined in parallel being equal to the sum of the reciprocals of the individual resistances.
Two checks turn this into a five-second question. The first is the bound — a parallel equivalent is always less than the smallest branch, and a series equivalent is always more than the largest. The second is the special case worth memorising: n equal resistors of value R in parallel give R/n, so three 33 Ω resistors would give 11 Ω. Household wiring is the practical reason the parallel rule matters. Every appliance is connected in parallel across the mains so that each receives the full supply voltage and can be switched independently; wire them in series and the voltage would divide between them and one failure would kill the whole circuit.
- For resistors in parallel, 1/Rp = 1/R1 + 1/R2 + 1/R3; for resistors in series, Rs = R1 + R2 + R3.
- A parallel combination always has an equivalent resistance smaller than its smallest branch; a series combination always larger than its largest.
- n equal resistors of value R in parallel give an equivalent resistance of R/n.
- In parallel the potential difference across each resistor is the same and the currents add; in series the current is the same and the potential differences add.
- Domestic wiring is a parallel arrangement so that each appliance gets the full mains voltage and can be operated independently.
- Applying the series rule to a parallel arrangement, which produces the largest option in the set.
- Forgetting the final reciprocal after summing the fractions, which gives 1/6 instead of 6.
- Ignoring the sanity bound; three of the four options here can be rejected without any calculation.
As a direct numerical on an equivalent resistance, as a wire-cut-into-equal-parts variation, or as a conceptual item on how the total changes when a resistor is added in parallel.
Domestic electrical wiring is basically a
- (a) series connection
- (b) parallel connection
- (c) combination of series and parallel connections
- (d) series connection within each room and parallel connection elsewhere
Answer(b) parallel connection
The same rule seen where it actually matters. Appliances are wired in parallel so each gets the full mains voltage and can be switched on its own, and the drop in equivalent resistance the CDS item calculates is what a house full of parallel loads produces.
Three equal resistors are connected in parallel configuration in a closed electrical circuit. Then the total resistance in the circuit becomes
- (a) one-third of the individual resistance.
- (b) two-third of the individual resistance.
- (c) equal to the individual resistance.
- (d) three times of the individual resistance.
Answer(a) one-third of the individual resistance.
The equal-resistor special case of the same formula. Three identical resistors in parallel give one-third of the individual value, which is the shortcut worth carrying into any question of this shape.
An electric wire of resistance 50 ohm is cut into five equal wires. These wires are then connected in parallel. What is the equivalent resistance of this combination?
- (a) 2 ohm
- (b) 10 ohm
- (c) 0·5 ohm
- (d) 5 ohm
Answer(a) 2 ohm
The same rule with a step in front of it. Cutting the wire into five gives five 10 ohm pieces, and five equal resistors in parallel give 10 divided by 5, which is 2 ohm.
- practice — not a real PYQ
Four resistors, each of 20 Ω, are connected in parallel. What is the equivalent resistance of the combination?
- (a)80 Ω
- (b)20 Ω
- (c)10 Ω
- (d)5 Ω
Answer(d) 5 Ω — n equal resistors of value R in parallel give R/n, so 20 divided by 4 is 5.
- practice — not a real PYQ
A 6 Ω and a 3 Ω resistor are connected in parallel. The equivalent resistance of the pair is
- (a)9 Ω
- (b)4.5 Ω
- (c)2 Ω
- (d)1.5 Ω
Answer(c) 2 Ω — the reciprocals give 1/6 + 1/3 = 1/2, so the equivalent is 2 Ω, less than the smaller of the two branches.