In an experiment, the interior of a first current-carrying toroid with 1000 turns per unit length is filled with aluminum (susceptibility 2.4 10⁻⁵ ). A second, identical toroid with 500 turns per unit length is filled with magnesium (susceptibility 1.2 10⁻⁵ ). If both toroids carry the same current I , the ratio of the increase in their magnetic fields is

Step-by-step solution

The magnetic field inside a toroid with a core is given by B = ₀ (1 + ) n I . The magnetic field without the core is B₀ = ₀ n I . The increase in the magnetic field due to the core is B = B - B₀ = ₀ n I . For the first toroid: n₁ = 1000 ₁ = 2.4 10⁻⁵ B₁ = ₀ ₁ n₁ I For the second toroid: n₂ = 500 ₂ = 1.2 10⁻⁵ B₂ = ₀ ₂ n₂ I The ratio of the increase in their magnetic fields is: B₁ B₂ = ₁ n₁ ₂ n₂ = 2.4 10⁻⁵ 1000 1.2 10⁻⁵ 500 = 2 2 = 4 The ratio is 4:1 . Answer: 4:1