A circular loop of radius r is formed by joining two semicircular wires of the same material but different cross-sectional areas A₁ and A₂ (where A₁ > A₂ ). A steady current I₀ enters the loop at one junction and leaves at the diametrically opposite junction. If the magnitude of the net magnetic field at the center of the loop is ₀ I₀ 8r , the ratio of their cross-sectional areas A₁ : A₂ is:

Step-by-step solution

Let the currents in the two semicircular branches be I₁ and I₂ , with I₁ flowing through the branch of area A₁ and I₂ through the branch of area A₂ . The magnetic field at the center of a semicircle carrying current I is B = ₀ I 4r . Since the currents flow in opposite angular directions, their magnetic fields at the center oppose each other. The net magnetic field is: B_ net = ₀ I₁ 4r - ₀ I₂ 4r = ₀ 4r (I₁ - I₂) Given B_ net = ₀ I₀ 8r , we have: ₀ 4r (I₁ - I₂) = ₀ I₀ 8r I₁ - I₂ = I₀ 2 Applying Kirchhoff's current l