A small circular loop of wire of radius r is placed inside a large regular hexagonal loop of wire of side a ( r a ). Both loops are coplanar and their centers coincide. The mutual inductance of the system is

Step-by-step solution

Let a current I flow in the large hexagonal loop. The magnetic field at the center is the vector sum of the magnetic fields produced by its six sides. The perpendicular distance from the center to any side of the regular hexagon is d = a (30^ ) = a 3 2 . The magnetic field due to one side of the hexagon at the center is: B₁ = ₀ I 4 d ( 30^ + 30^ ) = ₀ I 4 ( a 3 2 ) ( 1 2 + 1 2 ) = ₀ I 2 a 3 Total magnetic field at the center: B = 6 B₁ = 6 ₀ I 2 a 3 = 3 ₀ I a Since r a , the magnetic field can be considered uniform