JEE Main 2022 · 29 July · morning · Single correct · +4 / −1
Two metallic wires of identical dimensions are connected in series. If σ₁ and σ₂ are the conductivities of the these wires respectively, the effective conductivity of the combination is :
- (a) (σ₁ σ₂)/(σ₁+σ₂)
- (b) (2 σ₁ σ₂)/(σ₁+σ₂)
- (c) (σ₁+σ₂)/(2 σ₁ σ₂)
- (d) (σ₁+σ₂)/(σ₁ σ₂)
Answer and solution
Answer (b) (2 σ₁ σ₂)/(σ₁+σ₂)
Solution
Physics sir · 3 steps
Series means resistances add, but the length doubles too. Do not forget that 2l in the combined wire.
- Each wire. Each wire has length l and area A. With R = l/(σ A), the resistances are R₁ = l/(σ₁ A) and R₂ = l/(σ₂ A).
- Combined wire. In series the pair acts like one wire of length 2l and the same area A. If its effective conductivity is σ, its resistance is 2l/(σ A), and this must equal R₁ + R₂.
2l/(σ A) = l/(σ₁ A) + l/(σ₂ A) ⇒ 2/σ = (σ₁ + σ₂)/(σ₁ σ₂)
- So. σ = (2σ₁ σ₂)/(σ₁ + σ₂), which is option (b).
The idea to keep. For an effective property of a combination, equate the total resistance to that of one wire with the combined dimensions. In series the length adds, in parallel the area adds.
Answer from the NTA final answer key. Solution by PrepWiser.