An equilateral triangular loop of side length 0.9 ~m carries a steady current of 5 ~A . The magnitude of the magnetic field at the centroid of the triangle 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 the equilateral triangle be a = 0.9 ~m and the current be I = 5 ~A . The perpendicular distance d from the centroid to any side of the triangle is: d = a 2 3 The angle subtended by each half of the side at the centroid is ₁ = ₂ = 60^ . The magnetic field due to one side of the triangle at the centroid is: B₁ = ₀ I 4 d ( ₁ + ₂) B₁ = 10⁻⁷ 5 0.9 2 3 ( 60^ + 60^ ) B₁ = 10⁻⁷ 10 3 0.9 3 = 30 10⁻⁷ 0.9 = 100 3 10⁻⁷ ~T Since the magnetic fields due to all three sides add up at the centroid, the net ma