A square loop of edge a constructed from a uniform wire is shown in the figure. A current i enters the loop at point A and exits at point C . Suppose instead that the current i enters the loop at point A and exits at point B . Determine the magnetic field at the centre of the loop.

Step-by-step solution

Let the resistance of each side of the square loop be r . The current i enters at A and exits at B . It divides into two parallel paths: the direct path AB and the path ADCB . The resistance of path AB is r , and the resistance of path ADCB is 3r . The current i₁ through AB is i₁ = i ( 3r r + 3r ) = 3i 4 . The current i₂ through ADCB is i₂ = i ( r r + 3r ) = i 4 . The magnetic field at the centre of the square due to a single wire of length a carrying current I is B = ₀ I 4 (a/2) ( 45^ + 45^ ) = 2 ₀ I a . The magne