Ampere Law Applications

Ampere's Circuital Law becomes most powerful when applied to current geometries with inherent symmetry. The four canonical applications — infinite wire, solenoid, toroid, and coaxial cable — cover nearly every GATE and university examination problem in magnetostatics. Each geometry requires a carefully chosen Amperian path that exploits symmetry to reduce the line integral to a simple algebraic equation for H.

Application 1 — Infinite Straight Wire

For a long straight wire carrying current I, a circular Amperian loop of radius r centered on the wire is the natural choice. By symmetry, H is constant in magnitude and tangential (aφ direction) everywhere on this circle. Therefore ∮H·dL = H·2πr = I, giving H = I/(2πr) A/m. The field is purely tangential, forming concentric circles around the wire.

Application 2 — Infinite Solenoid

An ideal solenoid has n turns per meter wound tightly around a cylindrical former. A rectangular Amperian loop is chosen with one horizontal side of length L inside the solenoid and the other horizontal side far outside where H ≈ 0. The two vertical sides contribute nothing because H is axial and dL is radial there (dot product zero). The result is H·L = n·L·I, giving H = nI inside, and H = 0 outside. The field inside is uniform and axial — this is the fundamental result used in inductor design.

The confinement of field inside the solenoid is an idealization that holds perfectly only for an infinite or very long solenoid. Near the ends, fringing fields exist. For a finite solenoid of N total turns and length l, n = N/l.

Application 3 — Toroid

A toroid is a solenoid bent into a doughnut (torus) shape with N total turns. A circular Amperian loop of radius r inside the toroidal core (mean radius r) encloses all N turns: H·2πr = N·I, giving H = NI/(2πr). A circular loop outside the toroid encloses equal forward and return currents, so I_enc = 0 and H = 0 outside. This makes the toroid an extremely efficient magnetic circuit — virtually no external field leakage.

The field inside a toroid is not perfectly uniform — it varies as 1/r, being stronger at the inner radius and weaker at the outer radius. For a thin toroid (cross-section much smaller than mean radius), H ≈ NI/(2πr_mean) is used as an approximation.

Application 4 — Coaxial Cable

A coaxial cable has an inner conductor of radius a (carrying current I) and an outer conductor of inner radius b, outer radius c (carrying return current I). Three regions are analyzed using three concentric circular Amperian loops:

Region 1 (r < a, inside inner conductor): The enclosed current is only the fraction of I passing through the area up to r. If current is uniformly distributed, I_enc = I·(πr²/πa²) = Ir²/a². Then H·2πr = Ir²/a², giving H = Ir/(2πa²). Field increases linearly with r inside the conductor.

Region 2 (a < r < b, the dielectric gap): I_enc = I (full inner conductor current). H = I/(2πr) — same as an isolated infinite wire. This is where the useful transmission line field exists.

Region 3 (r > c, outside both conductors): I_enc = I (inner) + (-I) (outer return) = 0. Therefore H = 0. This is why coaxial cables are self-shielding — no external magnetic field.

Given:
A toroid has N = 500 turns, mean radius r = 0.1 m, carries I = 4 A.
Find H inside the toroid core.

Why this formula applies:
Circular Amperian loop of radius r inside toroid core encloses all N turns.
∮H·dL = H × 2πr = N × I

Formula:
H = NI / (2πr)

Substitution:
H = (500 × 4) / (2π × 0.1)
H = 2000 / 0.6283

Calculation:
H = 3183 A/m

Final Answer:
H ≈ 3183 A/m  inside the toroid core (tangential, in aφ direction)

Exam Tip: For a toroid, H is nonzero only inside the core — zero outside. For a solenoid, H = nI inside and zero outside. For a coaxial cable, the three regions give H ∝ r (inside inner), H ∝ 1/r (in gap), and H = 0 (outside both). Memorize these three profiles — they are directly tested in GATE numerical questions.

Comparison of Field Profiles

• Infinite wire: circular Amperian loop, H = I/(2πr), field extends to infinity with 1/r decay.
• Solenoid: rectangular loop, H = nI inside (uniform), H = 0 outside. Field is fully confined in ideal case.
• Toroid: circular loop inside core, H = NI/(2πr). Zero external field — ideal for EMI-sensitive applications.
• Coaxial cable: three regions — H rises linearly inside inner conductor, falls as 1/r in dielectric gap, drops to zero outside outer conductor.
• In all cases, the key step is choosing an Amperian path where H is constant and parallel to dL, then applying H × path_length = I_enc.

Quick Revision

• Infinite wire: H = I/(2πr), circular Amperian loop, field in aφ direction.
• Solenoid: H = nI inside (n = turns/m), H = 0 outside. Rectangular loop used.
• Toroid: H = NI/(2πr) inside core, H = 0 outside. N = total turns.
• Coaxial cable: H = Ir/(2πa²) for r < a; H = I/(2πr) for a < r < b; H = 0 for r > c.
• Exam trap: Inside a solid conductor (not hollow), H increases linearly with r, not constant and not zero.
• Toroid has zero external leakage field — solenoid has fringing fields near ends.
• All four problems use ∮H·dL = I_enc — the geometry of the Amperian path and the fraction of enclosed current are the only things that change.