Two resistors R₁ and R₂ arranged in parallel combination in an electrical closed circuit are made of the same material and of same thickness. If the length of R₂ is twice the length of R₁, then the total resistance R satisfies
- (a)3R = 2R₁
- (b)3R = 2R₂
- (c)2R = 3R₁
- (d)2R = 3R₂
Correct — A, 3R = 2R₁. Resistance is proportional to length for wires of the same material and thickness, so R₂ = 2R₁ because R₂ is twice as long. For two resistors in parallel, R = (R₁·R₂)/(R₁+R₂) = (R₁·2R₁)/(R₁+2R₁) = 2R₁²/3R₁ = 2R₁/3. Multiplying through by 3 gives 3R = 2R₁, which is option (a).
- (b)3R = 2R₂ — Since R₂ = 2R₁, the correct relation 3R = 2R₁ becomes 3R = R₂, not 2R₂, so this is wrong.
- (c)2R = 3R₁ — This would make R = 1.5R₁, but a parallel combination is smaller than either branch: R = 2R₁/3 ≈ 0.67R₁, so 2R = 3R₁ is incorrect.
- (d)2R = 3R₂ — This makes R even larger (1.5R₂ = 3R₁), whereas the true parallel resistance R = 2R₁/3 is smaller than both resistors.
For a uniform wire, resistance R = ρL/A, so with the same material (ρ) and thickness (A) the resistance is proportional to length. Two resistors in parallel combine as 1/R = 1/R₁ + 1/R₂, giving an equivalent resistance smaller than either branch.
First convert the length ratio into a resistance ratio (R₂ = 2R₁), then substitute into the parallel formula. The result, R = 2R₁/3, rearranges directly to 3R = 2R₁.
- For a uniform wire, R = ρL/A, so R is proportional to length when material and cross-section are fixed.
- R₂ is twice as long as R₁, so R₂ = 2R₁.
- In a parallel combination, R = R₁R₂/(R₁+R₂).
- The equivalent parallel resistance is always less than the smallest branch resistance.
Substitute R₂ = 2R₁ into the parallel formula to get 3R = 2R₁.
- Forgetting R₂ = 2R₁ from the length ratio.
- Expecting the parallel value to be larger than a branch — it is always smaller.
Combine the length–resistance relation (R₂ = 2R₁) with the parallel formula to reach 3R = 2R₁.
No directly related past PYQ was found.
- practice — not a real PYQ
Two wires of the same material and thickness have lengths in the ratio 1 : 2. The ratio of their resistances is
- (a)2 : 1
- (b)1 : 1
- (c)1 : 2
- (d)4 : 1
Answer(c) 1 : 2 — resistance is proportional to length.
- practice — not a real PYQ
The equivalent resistance of resistors connected in parallel is always
- (a)greater than the largest resistor
- (b)equal to their sum
- (c)less than the smallest resistor
- (d)zero
Answer(c) less than the smallest resistor.