A student wants to use Ampere's law to find the magnetic field at distance r from a long straight wire. They draw a square Amperian loop of side length r centered on the wire. Can they solve for B?
AYes — any closed loop gives the same result; the choice of loop shape doesn't matter
BNo — on a square loop, B varies in magnitude and direction along each side, so the integral B·dL cannot be simplified to B times a simple length
CNo — Ampere's law only applies to circular loops by definition
DYes — since I_enclosed is the same for any loop enclosing the wire, B can be computed directly
Ampere's law ∮B·dL = μ₀I_enclosed is always *true* for any closed loop, but it's only *useful* when you can factor B out of the integral. On a square loop around a wire, B changes in magnitude and direction at every point — the integral is a complicated mess that doesn't simplify without already knowing B. A circular loop of radius r works because cylindrical symmetry guarantees B is tangential and constant in magnitude everywhere on the circle, turning the integral into B(2πr). Option D correctly notes I_enclosed is independent of loop shape but wrongly concludes B can be found — you need to evaluate the left side, not just the right.
Question 2 Multiple Choice
The result B = μ₀nI inside a solenoid (where n is turns per unit length) comes from applying Ampere's law with a rectangular loop. Which feature of that loop analysis produces this result?
AThe rectangular loop must enclose all N turns of the solenoid to capture the total current
BA rectangular loop straddling the solenoid wall has only one side contributing to ∮B·dL — the segment inside — because B ≈ 0 outside and B is perpendicular to the two transverse sides
CThe circular winding geometry of the solenoid ensures B is constant everywhere on any rectangular loop
DBiot-Savart must first confirm the field is uniform before Ampere's law can be applied
The rectangle is chosen to straddle the solenoid wall (one segment inside, one outside, two transverse segments). Outside: B ≈ 0, contributing nothing. Transverse sides: B is parallel to the solenoid axis, perpendicular to dL along those sides, contributing nothing. Inside: B is parallel to dL and uniform, contributing B × L. The right side is μ₀ × (nL) × I, since nL turns pass through the loop. Thus B × L = μ₀nLI → B = μ₀nI. The loop doesn't need to enclose all turns (option A); it just needs to enclose a known number of them.
Question 3 True / False
Ampere's law ∮B·dL = μ₀I_enclosed is always mathematically valid, but it is only practically useful for computing magnetic fields when the current distribution has sufficient symmetry.
TTrue
FFalse
Answer: True
Exactly right. Ampere's law is an exact statement about any closed loop, but it becomes a useful computational tool only when you can choose an Amperian loop where B is constant and parallel on part of the loop (and zero or perpendicular on the rest). Without that symmetry, the law gives you a true equation with an unknown integral on the left — unsolvable without already knowing B everywhere. Biot-Savart, though harder to compute, works for asymmetric configurations where Ampere's law cannot be usefully applied.
Question 4 True / False
The Amperian loop used in Ampere's law is expected to correspond to an actual physical conductor or current path in the problem.
TTrue
FFalse
Answer: False
False — the Amperian loop is a purely mathematical construct chosen for computational convenience. It has no physical reality and need not correspond to any conductor or circuit. For a solenoid, we choose a rectangle that straddles the solenoid wall; for a wire, we choose a circle around it. These are imaginary geometric objects drawn to exploit symmetry. The only physical requirement is that I_enclosed correctly counts the net current passing through any surface bounded by the loop.
Question 5 Short Answer
What is the key criterion for choosing a useful Amperian loop, and why does that criterion matter for the calculation?
Think about your answer, then reveal below.
Model answer: The Amperian loop should be chosen so that B is either (1) parallel to dL and constant in magnitude — so ∮B·dL = B × (loop length) — or (2) perpendicular to dL — contributing zero. Without this, the left side of Ampere's law ∮B·dL is a complicated integral that cannot be evaluated without already knowing B in detail, defeating the purpose. Symmetry is what makes such a loop choice possible: cylindrical symmetry for a wire (circular loop), translational symmetry for a solenoid (rectangular loop), and toroidal symmetry for a toroid (circular loop inside the coils).
This is the direct analog of choosing a Gaussian surface in Gauss's law: both techniques work by turning a hard integral into a trivial product using the fact that the field is uniform and aligned on the chosen surface or loop. The underlying law is always valid; the choice of integration path or surface determines whether it yields a usable equation.