Two identical square loops, each of side length 20 ~cm , are placed such that they share a common centre. One loop lies entirely in the xy -plane and the other lies entirely in the yz -plane. Both loops carry the same steady current of 5 ~A . The magnitude of the net magnetic field at their common centre is x 10⁻⁶ ~T . The value of x is ______. [ Take ₀ = 4 10⁻⁷ ~T~m~A ⁻¹ ]

Step-by-step solution

Let the side length of each square loop be a = 0.2 ~m and the current be I = 5 ~A . First, we find the magnetic field at the centre of a single square loop. The perpendicular distance from the centre to any side is d = a 2 = 0.1 ~m . The angle subtended by each half of the side at the centre is ₁ = ₂ = 45^ . The magnetic field due to one side at the centre is: B_ side = ₀ I 4 d ( 45^ + 45^ ) B_ side = 10⁻⁷ 5 0.1 ( 1 2 + 1 2 ) = 50 2 10⁻⁷ ~T The magnetic field due to the entire square loop is: B_ sq = 4 B_ side = 20