Which statement about the magnetic field of an ideal toroid is true?

Which statement about the magnetic field of an ideal toroid is true?

• A. Outside the toroid, the magnetic field is approximately zero
• B. Outside the toroid, the magnetic field is twice the inside field
• C. Outside the toroid, the magnetic field equals the inside field
• D. The magnetic field is undefined outside the toroid

Answer: (A)

A fundamental idea here is that an ideal toroid confines its magnetic field inside the doughnut-shaped core. By Ampère’s law, the line integral of B around any loop that encircles the torus depends on the current it threads. A loop drawn outside the toroid encloses no net current, so ∮B·dl = μ0 I_enc = 0. With the symmetry of the winding, this forces the magnetic field outside to be essentially zero. Inside the toroid, the field is azimuthal and nonzero, given roughly by B ≈ μ0 N I /(2π r) at radius r within the core. So the statement that the magnetic field outside the toroid is approximately zero is the correct description. The other options would imply a nonzero external field or a relationship that doesn’t hold for an ideal toroid, and in the ideal case the outside field is not defined or simply not present. Real toroids may have tiny leakage fields, but those are usually negligible.

A fundamental idea here is that an ideal toroid confines its magnetic field inside the doughnut-shaped core. By Ampère’s law, the line integral of B around any loop that encircles the torus depends on the current it threads. A loop drawn outside the toroid encloses no net current, so ∮B·dl = μ0 I_enc = 0. With the symmetry of the winding, this forces the magnetic field outside to be essentially zero. Inside the toroid, the field is azimuthal and nonzero, given roughly by B ≈ μ0 N I /(2π r) at radius r within the core. So the statement that the magnetic field outside the toroid is approximately zero is the correct description. The other options would imply a nonzero external field or a relationship that doesn’t hold for an ideal toroid, and in the ideal case the outside field is not defined or simply not present. Real toroids may have tiny leakage fields, but those are usually negligible.