A uniform conducting wire of total length 60L is used to form two separate current-carrying coils. First, the entire wire is wound to form a square coil of side 3L . Later, the same wire is unwound and completely rewound to form a rectangular coil of length 4L and width L . If both coils carry the same steady current I , what is the ratio of the magnetic dipole moment of the square coil to that of the rectangular coi

Step-by-step solution

Total length of the wire, L_ total = 60L . For the square coil: Side of the square, a = 3L . Perimeter of one turn = 4a = 12L . Number of turns, N₁ = 60L 12L = 5 . Area of one turn, A₁ = a^2 = 9L^2 . Magnetic dipole moment, M₁ = N₁ I A₁ = 5 I 9L^2 = 45 I L^2 . For the rectangular coil: Length = 4L , width = L . Perimeter of one turn = 2(4L + L) = 10L . Number of turns, N₂ = 60L 10L = 6 . Area of one turn, A₂ = 4L L = 4L^2 . Magnetic dipole moment, M₂ = N₂ I A₂ = 6 I 4L^2 = 24 I L^2 . Ratio of the magnetic dipole mo