A circular loop of radius r is formed by joining two semicircular wires of the same material. The cross-sectional area of the first semicircular wire is A and that of the second is 2A . A total steady current I enters the loop at one junction and leaves at the other. The magnitude of the net magnetic field at the center of the loop is:

Step-by-step solution

Let the resistance of the first semicircular wire be R₁ and the second be R₂ . The resistance of a wire is given by R = l A . Since both are semicircles of radius r , their lengths are equal ( l = r ). R₁ = r A and R₂ = r 2A = R₁ 2 The two semicircular wires are connected in parallel between the junctions, so the potential difference across them is the same. I₁ R₁ = I₂ R₂ I₁ R₁ = I₂ ( R₁ 2 ) I₂ = 2I₁ The total current is I = I₁ + I₂ = I₁ + 2I₁ = 3I₁ . Thus, I₁ = I 3 and I₂ = 2I 3 . The magnetic field at the center