Two infinitely long wires are placed perpendicular to the plane of paper. Current in wire A to 4 i0 outward the plane of paper and current in ‘B’ is i0 inward the plane of paper. The ∫− α+ αB→⋅ dℓ→ along the QP is K(μ0i0). Find the value of K.

Imagine a rectangular Amperian loop (cdefc) of very long length and very small breadth (almost zero).For f→c : ∫B→⋅dℓ→=x (say)For c→d : ∫B→⋅dℓ→=0 (Magnetic field is zero for large distances)For d→e : ∫B→⋅dℓ→=x (symmetry)For e→f : ∫B→⋅dℓ→=0 (Magnetic field is zero for large distances)∴For loop cdefc, using Ampere's Law, ∮B→⋅dℓ→=μ0ienclosed⇒x+0+x+0=μ04i0⇒x=μ02i0⇒∫fcB→⋅dℓ→=2μ0i0Now, Imagine another rectangular Amperian loop (rstur) of very long length and very small breadth (almost zero).For r→s : ∫B→⋅dℓ→=x (say)For s→t : ∫B→⋅dℓ→=0 (Magnetic field is zero for large distances)For t→u : ∫B→⋅dℓ→=x (symmetry)For u→r : ∫B→⋅dℓ→=0 (Magnetic field is zero for large distances)∴For loop rstur, using Ampere's Law, ∮B→⋅dℓ→=μ0ienclosed⇒x+0+x+0=μ0i0 ⇒x=μ00.5i0⇒∫rsB→⋅dℓ→=0.5μ0i0Since, ∫fcB→⋅dℓ→ and ∫rsB→⋅dℓ→ are in same direction, we can add them∴∫-α+αB→⋅dℓ→=∫fcB→⋅dℓ→+∫rsB→⋅dℓ→ =2μ0i0+0.5μ0i0=2.5μ0i0