The magnitude of the current in the loop is 40.8 A and the direction of the current in the loop is such that it produces a magnetic field that is in the opposite direction of the magnetic field produced by the solenoid.
To determine the magnitude and direction of the current that makes the magnetic field at the center of the solenoid zero, we can use the principle of superposition. The magnetic field at the center of the solenoid is the sum of the magnetic fields produced by each of the turns of the solenoid. The magnetic field produced by a single turn is given by Ampère's law:
B = μ₀I/2πa
where:
B is the magnetic field
μ₀ is the permeability of free space
I is the current in the turn
a is the distance from the center of the turn to the point where the magnetic field is being measured
In this case, I = 22.0 mA, a = 7.50 cm, and μ₀ = 4π × 10⁻⁷ T⋅m/A. The magnetic field produced by a single turn is therefore:
B = (4π × 10⁻⁷ T⋅m/A) × (22.0 × 10⁻³ A) / (2π × 0.0750 m) ≈ 1.47 × 10⁻⁴ T
The magnetic field produced by each turn is in the same direction, so the magnetic fields produced by all of the turns add up. The total magnetic field at the center of the solenoid is therefore:
B_total = N × B = 350 × 1.47 × 10⁻⁴ T ≈ 0.515 T
We want the magnetic field at the center of the solenoid to be zero, so we need to add a second magnetic field that cancels out the first one. The magnetic field produced by a single loop is given by:
B = μ₀NI/2πr
where:
B is the magnetic field
μ₀ is the permeability of free space
N is the number of turns in the loop
I is the current in the loop
r is the distance from the center of the loop to the point where the magnetic field is being measured
In this case, N = 1, r = 10.0 cm, and μ₀ = 4π × 10⁻⁷ T⋅m/A. The magnetic field produced by the loop is therefore:
B = (4π × 10⁻⁷ T⋅m/A) × 1 × I / (2π × 0.100 m) ≈ 2.00 × 10⁻⁵ T
The magnetic field produced by the loop is in the opposite direction of the magnetic field produced by the solenoid, so we can add the two magnetic fields to get the total magnetic field at the center of the solenoid. The total magnetic field is therefore:
B_total = B_solenoid + B_loop = 0.515 T - 2.00 × 10⁻⁵ T ≈ 0.513 T
The magnitude of the current in the loop is therefore:
I = (2πr/μ₀) × B × N ≈ (2π × 0.100 m / (4π × 10⁻⁷ T⋅m/A)) × 0.513 T × 1 ≈ 40.8 A