For an ideal toroid with N turns and mean radius r, which statement correctly describes the magnetic field inside the core?

For an ideal toroid with N turns and mean radius r, which statement correctly describes the magnetic field inside the core?

• A. B = μ0 N I /(2π r) inside; outside B ≈ 0
• B. B = μ0 I /(2π r) inside; outside B ≈ 0
• C. B = μ0 N I /(2π r^2) inside; outside B ≈ 0
• D. B = 0 inside; outside B = μ0 N I /(2π r)

Answer: (A)

Ampere's law shows that for an ideal toroid the magnetic field is confined to the core and is set by the total current linking the coil. Consider a circular Amperian path of radius r that lies inside the toroid and loops once around the central axis. Each of the N turns carries current I, so the enclosed current is N I. The line integral around this path is ∮ B · dl = B · (2π r) = μ0 N I, giving the field inside as B = μ0 N I /(2π r). This 1/r dependence and the N factor come directly from Ampere's law and the fact that there are N linked turns. Outside the toroid, a loop around the torus encloses no net current because the currents loop inside the donut; thus ∮ B · dl = 0, and for an ideal toroid the magnetic field there is essentially zero. The field direction inside is azimuthal, circling the core, consistent with the right-hand rule applied to each turn. Real toroids may have tiny leakage, but in the ideal case the outside field is negligible.

Ampere's law shows that for an ideal toroid the magnetic field is confined to the core and is set by the total current linking the coil. Consider a circular Amperian path of radius r that lies inside the toroid and loops once around the central axis. Each of the N turns carries current I, so the enclosed current is N I. The line integral around this path is ∮ B · dl = B · (2π r) = μ0 N I, giving the field inside as B = μ0 N I /(2π r). This 1/r dependence and the N factor come directly from Ampere's law and the fact that there are N linked turns.

Outside the toroid, a loop around the torus encloses no net current because the currents loop inside the donut; thus ∮ B · dl = 0, and for an ideal toroid the magnetic field there is essentially zero. The field direction inside is azimuthal, circling the core, consistent with the right-hand rule applied to each turn. Real toroids may have tiny leakage, but in the ideal case the outside field is negligible.