An Amperian loop is drawn as a circle with a radius larger than the outer radius of a toroid carrying current. What is the magnetic field along this loop?
AB = μ₀NI/(2πr), the same formula as inside the toroid
BB is nonzero but weaker than inside, because you are farther from the wires
CB = 0, because the net enclosed current is zero
DB is undefined outside the toroid geometry
For any Amperian loop outside the toroid, every wire carrying current in one direction around the ring is paired with a return wire carrying current in the opposite direction. All N turns thread through the loop twice — once in each direction — so the net enclosed current is exactly zero. Ampere's law then gives B(2πr) = μ₀(0) = 0. This perfect cancellation is why toroids confine their field completely, unlike solenoids whose field lines loop back through external space.
Question 2 Multiple Choice
Compared to a solenoid with the same number of turns and current, what is unique about the field distribution inside a toroid?
AThe toroid's interior field is uniform, just like a solenoid's, because both use the same winding technique
BThe toroid's interior field is zero because bending the solenoid cancels the contributions
CThe toroid's interior field varies as 1/r — stronger near the inner radius, weaker near the outer radius
DThe toroid's interior field is stronger everywhere than the equivalent solenoid field
Applying Ampere's law to a circular loop of radius r inside the toroid gives B(2πr) = μ₀NI, so B = μ₀NI/(2πr). The 1/r dependence means the field is not uniform: it is strongest at the inner radius and falls off toward the outer radius. This contrasts with a straight solenoid, where the interior field B ≈ μ₀nI is uniform everywhere. The nonuniformity arises directly from the different path lengths (2πr) of the Amperian loops at different radii.
Question 3 True / False
The magnetic field inside a toroid is uniform, just as it is inside a long solenoid.
TTrue
FFalse
Answer: False
This is false. While a solenoid has an approximately uniform field B = μ₀nI throughout its interior, the toroid's field varies inversely with radius: B(r) = μ₀NI/(2πr). Points closer to the inner radius see a stronger field than points near the outer radius. The curvature introduced by bending the solenoid into a ring changes the Amperian loop lengths, breaking the uniformity that a straight solenoid provides.
Question 4 True / False
The magnetic field outside a toroid is exactly zero everywhere, with no external flux leakage.
TTrue
FFalse
Answer: True
True. For any Amperian loop drawn entirely outside the toroid, the full winding passes through it in both directions, making the net enclosed current zero. By Ampere's law, B = 0. This is the fundamental advantage of a toroidal geometry: complete field containment, with no electromagnetic coupling to neighboring circuit elements. This is why toroids are preferred in power electronics and audio circuits where interference between components must be minimized.
Question 5 Short Answer
Using Ampere's law, explain why the magnetic field outside a toroid is exactly zero, even though current is flowing through all N turns of the winding.
Think about your answer, then reveal below.
Model answer: The winding wraps all the way around the toroid's 360° ring. Any Amperian loop drawn outside the toroid encloses every one of the N turns twice: once as the wire carries current in one direction, and once as it returns in the opposite direction. The positive and negative contributions cancel exactly, giving zero net enclosed current. Ampere's law (∮B·dl = μ₀I_enc) then requires B = 0 along that loop. This complete cancellation — which does not occur for a straight solenoid, where the external loops only thread the return path at the ends — is what makes the toroid's field confinement perfect.
Students often assume that 'more current' means 'more field everywhere,' but the direction of current matters as much as its magnitude. The toroidal geometry forces every outward-going current segment to be paired with an equal and opposite return segment inside any external Amperian loop, producing exact cancellation.