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 toroidal solenoid can be calculated using the formula B =μ₀NI/2πr, where N is the number of turns in the coil, I is the current flowing through the coil, and r is the radius of the toroid. Since we are given the magnetic field at the center of the solenoid, we can use this formula to find the current flowing through the coil.

2.0 μT = μ₀NI/2π(0.50 m)
I = 1.27 A

Now we can use the formula for magnetic field at a point on the axis of a solenoid, which is given by B =μ₀nI, where n is the number of turns per unit length of the solenoid. The maximum value of the magnetic field will occur at the edge of the cross section of the solenoid, so we need to find the number of turns per unit length at this point.

The circumference of the cross section is π(0.10 m) = 0.314 m. If we assume that all of the turns in the solenoid are evenly spaced along the length, then the number of turns per unit length at the edge of the cross section is N/(2πr), where N is the total number of turns in the solenoid.

N/(2πr) = n(0.314 m)
n = N/(2πr)(0.314 m)

Now we can use the formula for the maximum magnetic field:

B =μ₀nI =μ₀N/(2πr)(0.314 m)(1.27 A) = 2.21 μT

Therefore, the largest value of the magnetic field inside the solenoid when this current is flowing is 2.2 μT, choice C.

A Toroidal Solenoid Has a Central Radius of 0