Two wires of same length and thickness having specific resistances 6Ω cm and 3Ω cm respectively are connected in parallel. The effective resistivity is ρΩ cm. The value of ρ, to the nearest integer, is ____________.

Solution

Physics sir · 4 steps

Two wires side by side act like one wire of the same length but twice the cross-section.

1. Each wire. Take each wire to have length l and area A. Then R₁ = 6l/A and R₂ = 3l/A.
2. Parallel resistance. Rₚ = (R₁ R₂)/(R₁ + R₂) = 18/9 · l/A = 2l/A.
3. Equivalent wire. The combination behaves like a single wire of length l and area 2A. Its resistance must equal Rₚ.

   (ρ l)/2A = 2l/A ⇒ ρ = 4 Ω cm

4. So. The effective resistivity is ρ = 4 Ω cm, so the answer is 4.

The common slip: 2. Putting the two resistivities straight into the parallel formula gives (6 × 3)/(6 + 3) = 2. That ignores the doubled cross-section of the combination.

The idea to keep. Resistivity is a material property, so find the combined resistance first. Then match it to an equivalent wire with the right length and area: parallel adds areas, series adds lengths.

Answer from the NTA final answer key. Solution by PrepWiser.