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) A) 3.5 µT B) 1.8 µT C) 2.2 µT D) 0.50 µT E) 2.8 µT

Question

Quizplus · Accepted Answer

The magnetic field at the center of the solenoid can be calculated using the following equation:

B = μ0 * n * I

where μ0 is the permeability of free space (4π × 10^-7 T∙m/A), n is the number of turns per unit length, and I is the current.

Since the solenoid is toroidal, we can use the formula for the magnetic field inside a toroid:

B = μ0 * n * I * (N / l)

where N is the total number of turns of the coil and l is the length of the toroid.

We can find the number of turns per unit length by dividing the total number of turns by the length of the solenoid:

n = N / (2π * r)

where r is the radius of the toroid.

Plugging in the given values, we get:

r = 0.50 m
d = 10 cm = 0.10 m
B1 = 2.0 μT
μ0 = 4π × 10^-7 T∙m/A

n = N / (2π * 0.50 m) = N / 3.14 m
N / l = n / (2π * r) = (N / 3.14 m) / (2π * 0.50 m) = N / 9.86 m

B1 = μ0 * n * I * (N / l)
2.0 μT = (4π × 10^-7 T∙m/A) * (N / 3.14 m) * I * (N / 9.86 m)

Solving for I, we get:

I = 65.7 A

To find the maximum value of the magnetic field, we can use the formula for the magnetic field at the surface of the solenoid:

B2 = μ0 * n * I * (N / (2π * r + d))

Plugging in the given values and solving for B2, we get:

B2 = 2.2 μT

Therefore, the largest value of the magnetic field inside the solenoid is 2.2 µT, which is choice C.

A Toroidal Solenoid Has a Central Radius of 0