Toroids: A toroidal solenoid has a central radius of 0.50 m and a cross-sectional diameter of 10 cm. When a current passes through the coil of the solenoid, the magnetic field inside the solenoid at its CENTER has a magnitude of 2.0 μT. What is the largest value of the magnetic field inside the solenoid when this current is flowing? (μ₀ = 4π × 10^-7 T ∙ m/A)

Question

Quizplus · Accepted Answer

The magnetic field inside a solenoid can be calculated using the formula B = μS1U1N1S1U1I1S1S1P0, where μS1U1B10S1U1B0 is the permeability of free space, N is the number of turns, and I is the current. Since the question asks for the largest value of magnetic field, we can assume that it occurs at the inner radius of the solenoid.

The inner radius of the solenoid is given as 0.50 m, and the cross-sectional diameter is given as 10 cm = 0.10 m. Therefore, the length of the solenoid is L = 2πr = 2π(0.50) = 3.14 m, and the number of turns is N = (cross-sectional area)/(area per turn) = (π/4)(0.10)²/(π/4)(0.50)² = 1/25.

Substituting these values into the formula, we get B = (4π × 10S1U1P1-7S1S1P0 T ∙ m/A)(1/25)(I/L) = (4/5π)(I/L). We know that at the center of the solenoid, B = 2.0 µT, which gives us the equation 2.0 µT = (4/5π)(I/L) or I = (5/8π)(2.0 µT)(3.14 m)/(1/25) = 9.8 A.

To calculate the largest value of magnetic field, we need to maximize the current. Using the formula for the magnetic field, we get B = (4/5π)(I/L), which gives us B = (4/5π)(9.8 A)/(3.14 m) = 2.2 µT. Therefore, the answer is C.

Toroids: a Toroidal Solenoid Has a Central Radius of 0