Ampère's Law

What is Ampère's Law?

In magnetostatics, Ampère's Law states that the line integral of B around any closed Amperian loop equals μ₀ times the current threading that loop: ∮B·dl = μ₀I_enc; the full Maxwell-Ampère version adds a displacement-current term μ₀ε₀ dΦ_E/dt for time-varying fields. The power of the magnetostatic form lies in symmetry: for a long straight wire the integrand is constant around a circle of radius r, giving B(2πr) = μ₀I and therefore B = μ₀I/(2πr). Inside an ideal solenoid with n turns per meter, a rectangular Amperian loop yields B = μ₀nI (uniform, axially directed) while outside B ≈ 0. The simulation lets you switch between a bare wire, a coaxial cable, and a solenoid, position the Amperian loop with loopRadius, and watch ∮B·dl update numerically as the enclosed current changes.

Parameters explained

Current I(A)

Current I controls how much charge flow threads the magnetic geometry. In the Infinite Straight Wire preset, increasing Current raises the magnetic field everywhere in direct proportion, so doubling Current doubles B at the same Loop Radius. In the Solenoid (n=1000) preset, Current also scales the interior field linearly through B = μ₀nI. In the Toroid preset, Current multiplies the enclosed turn contribution, so the field inside the toroidal path grows while the outside remains much smaller in the ideal model. Keep Loop Radius and Turns fixed, then sweep Current to isolate this linear dependence.

Loop Radius(m)

Loop Radius sets the size of the Amperian loop used to sample circulation. For the Infinite Straight Wire preset, a larger Loop Radius means the same enclosed current is spread around a longer circular path, so B decreases as 1/r even though ∮B·dl remains tied to μ₀I_enc. In toroidal reasoning, radius also matters because the field around the core follows B = μ₀NI/(2πr) inside the winding region. Use this slider after choosing a preset to test whether changing the loop geometry changes enclosed current, path length, or both.

Turns(turns)

Turns controls the number of windings used by coil-based cases. It has little conceptual role for the Infinite Straight Wire preset, where the conductor is modeled as one long current path. In the Solenoid (n=1000) preset, more Turns means more current loops per length, increasing the interior field predicted by B = μ₀nI. In the Toroid preset, Turns represents the total winding count N enclosed by an Amperian loop inside the core, giving B = μ₀NI/(2πr). Keep Current fixed while changing Turns to see how repeated windings amplify magnetic circulation.

Common misconceptions

• MisconceptionAmpère's Law works for any closed loop — you can always use it to find B directly, even for a bent or irregular wire.

  CorrectMathematically ∮B·dl = μ₀I_enc holds for any closed loop, but B can only be factored out of the integral when the field has high symmetry (constant magnitude along the loop, constant angle with dl). For irregular geometries you cannot simplify the integral, so Ampère's Law gives a true result but not a useful one for calculating B.

• MisconceptionThe magnetic field inside a solenoid points in the same direction as the current in the wire.

  CorrectThe field inside a solenoid points along the solenoid axis, which is perpendicular to the current-carrying wire segments. Apply the right-hand rule: curl the fingers in the direction of current flow around the turns and the thumb points along the axis in the direction of B.

• MisconceptionThe magnetic field outside a solenoid is just weaker, not zero.

  CorrectFor an ideal infinite solenoid, B outside is zero by a symmetry argument: the field must be uniform and vanish at infinity for an infinite uniform solenoid. An Amperian rectangle straddling the wall with one long side inside and one outside gives B_inside − B_outside = μ₀nI, consistent with B_inside = μ₀nI when B_outside = 0. Real finite solenoids have a small but nonzero exterior field.

• MisconceptionA coaxial cable with equal and opposite inner and outer currents still produces a B field outside the cable.

  CorrectAny Amperian loop enclosing the full coaxial cable sees I_enc = +I − I = 0, so ∮B·dl = 0 and B = 0 outside. This is why coaxial cables are used in precision electronics — they confine the magnetic field entirely within the outer conductor.

• MisconceptionDoubling the loop radius doubles the magnetic field from a long wire because the loop encloses more area.

  CorrectB = μ₀I/(2πr) falls as 1/r — doubling r halves B. Ampère's Law relates the integral to I_enc, which does not change with loop size for a single wire. The field drops because the same current is spread over a larger circumference.

How teachers use this lab

1. Symmetry argument gateway: before running the simulation, ask students to write the general Ampère's Law ∮B·dl = μ₀I_enc and explain what additional physical argument lets them replace the integral with B(2πr). Use the Infinite Straight Wire preset first so Current and Loop Radius map directly onto B = μ₀I/(2πr).
2. Wire field data collection: choose the Infinite Straight Wire preset, hold Current constant, sweep Loop Radius from 0.1 m to 2.0 m, record the displayed B at each step, and plot B vs. 1/r. Students should obtain a straight line through the origin with slope μ₀I/(2π). Verify by extracting μ₀ = 2π × slope / I and comparing to the accepted 4π × 10⁻⁷ T·m/A.
3. Solenoid scaling lab: choose the Solenoid (n=1000) preset, hold Loop Radius fixed, and compare several Current values. Then hold Current fixed and sweep Turns from 100 to 2000. Students should identify the linear relationships in B = μ₀nI and explain why many windings make the interior field stronger.
4. Toroid enclosed-current challenge: choose the Toroid preset and ask students to predict how increasing Turns changes ∮B·dl. Then vary Loop Radius and discuss why the ideal toroid field is strongest inside the winding region and much smaller outside, even though each wire still carries the same Current.
5. Right-hand rule 3D practice: with the Infinite Straight Wire preset, increase Current to 10 A and ask students to verify the rotational direction of B vectors using the right-hand rule with thumb along the current direction. Then switch to Solenoid (n=1000) and Toroid presets so students compare circular field lines, axial interior fields, and confined toroidal fields.