Ampère's Law - AP Physics C Electricity and Magnetism

SECTION 1

Historical Context & Motivation

The connection between electricity and magnetism was one of the great intellectual triumphs of nineteenth-century physics. In 1820, Hans Christian Ørsted's accidental observation that a current-carrying wire deflected a nearby compass needle shattered the long-held belief that electric and magnetic phenomena were entirely separate. Within weeks, André-Marie Ampère launched a systematic investigation of the forces between current-carrying conductors. His work culminated in a mathematical relationship—now called Ampère's Law—that elegantly links the magnetic field circulating around a closed path to the net electric current threading through that path.

1820 • Ørsted's Discovery

Hans Christian Ørsted demonstrates that an electric current produces a magnetic field, linking electricity and magnetism for the first time.

1820–1825 • Ampère's Force Law

André-Marie Ampère quantifies the mutual forces between current-carrying wires and formulates the circuital relationship between current and the magnetic field.

1831 • Faraday's Induction

Michael Faraday discovers electromagnetic induction, revealing the reciprocal link: changing magnetic fields produce electric fields.

1861–1865 • Maxwell's Equations

James Clerk Maxwell adds the displacement-current term to Ampère's Law and unifies electricity, magnetism, and optics in four elegant equations.

The central question Ampère's Law addresses is deceptively simple: given a known distribution of steady currents, how can we efficiently determine the magnetic field they produce? While the Biot–Savart law answers this question in full generality—at the cost of a sometimes formidable vector integral—Ampère's Law exploits symmetry to collapse that integral into a single algebraic step, much as Gauss's law does for electric fields.

SECTION 2

Core Principles & Definitions

Ampère's Law is a statement about the line integral of the magnetic field around a closed loop, called an Amperian loop. The law holds for any closed path in space, but it becomes a powerful computational tool only when the loop is chosen to match the symmetry of the current distribution. Before diving into the mathematics, it is essential to internalize several foundational ideas.

Circulation of B

The line integral ∮ B · dl around a closed loop measures how much the magnetic field 'wraps' around the path. A nonzero circulation means net current passes through the loop.

Enclosed Current I_enc

Only currents that actually pierce the surface bounded by the Amperian loop contribute to I_enc. Currents outside the loop create fields along the path but their contributions cancel over a full traversal.

Right-Hand Rule

Curl the fingers of your right hand in the direction you traverse the loop; your thumb points in the direction of positive current. Currents flowing opposite to the thumb are counted as negative.

Symmetry Requirement

Ampère's Law is universally true, but it is useful for computing B only when the geometry allows you to factor B out of the integral—requiring cylindrical, planar, or toroidal symmetry.

SECTION 3

Visual Explanation

Magnetic Field Lines Around a Long Straight Wire

Two concentric Amperian loops of radii r₁ and r₂ surround a long straight wire carrying current I out of the page. By symmetry, B is tangent to each loop and constant in magnitude, so the line integral reduces to B × (2πr) = μ₀I.

The diagram above illustrates the canonical application of Ampère's Law: a long, straight wire carrying steady current I directed out of the page. The magnetic field lines form concentric circles centered on the wire, and the field's magnitude depends only on the radial distance r from the wire. When we choose a circular Amperian loop of radius r, B is everywhere tangent to the loop and has constant magnitude along it. The dot product B · dl therefore equals B dl at every point, and integrating over the full circumference gives B(2πr). Setting this equal to μ₀I_enc immediately yields B = μ₀I/(2πr), reproducing the Biot–Savart result in a single line.

SECTION 4

Mathematical Framework

AMPÈRE'S LAW (INTEGRAL FORM)

∮ B · dl = μ₀ I_enc

B = magnetic field (T), dl = infinitesimal displacement along the loop, μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space), I_enc = net current piercing the surface bounded by the loop.

The integral is a closed line integral: you traverse a complete loop and sum the component of B parallel to the path at each infinitesimal step. The sign of I_enc is determined by the right-hand rule: if you curl the fingers of your right hand in the direction of integration, your thumb defines the positive-current direction. Currents flowing opposite to your thumb contribute negatively.

LONG STRAIGHT WIRE

B = μ₀I / (2πr)

r = perpendicular distance from the wire. B is directed tangentially (circumferentially) around the wire per the right-hand rule.

IDEAL SOLENOID (INTERIOR)

B = μ₀ n I

n = N/L = number of turns per unit length, I = current through the solenoid. The field is uniform inside and approximately zero outside.

TOROID

B = μ₀NI / (2πr)

N = total number of turns, r = distance from the center of the toroid (inside the windings only). The field vanishes outside the toroid.

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Differential Form (for reference)

SECTION 5

Key Geometries & Applications

Ampère's Law becomes a practical tool when the current distribution possesses enough symmetry to make the magnetic field constant (or zero) along portions of the Amperian loop. The three classic geometries tested on the AP exam are the long straight wire, the ideal solenoid, and the toroid. A fourth important case—current distributed uniformly across the cross-section of a thick wire—combines the techniques of the first three.

A rectangular Amperian loop is used to derive the field inside an ideal solenoid. Only side a (inside, parallel to B) contributes; sides b and d are perpendicular to B, and side c lies outside where B ≈ 0.

Geometry	Amperian Loop Shape	Result for B	Key Insight
Long straight wire	Circle coaxial with wire, radius r	B = μ₀I / (2πr)	B ∝ 1/r; field is purely azimuthal
Ideal solenoid	Rectangle with one side inside, one outside	B = μ₀nI (inside); B ≈ 0 (outside)	Uniform interior field; n = N/L
Toroid	Circle inside toroidal windings	B = μ₀NI / (2πr)	B = 0 outside toroid; depends on r inside
Thick wire (r < R)	Circle inside wire cross-section	B = μ₀Ir / (2πR²)	B ∝ r inside; only fraction of I enclosed

Summary of standard Ampère's Law applications

SECTION 6

Worked Example

Worked example

1
Step 1 — Identify Given Values

A long cylindrical conductor of radius R = 3.0 cm carries a total current I = 12 A distributed uniformly across its cross-section. Find the magnetic field at a distance r = 1.0 cm from the center of the wire.
2
Step 2 — Choose the Amperian Loop

By symmetry, B is azimuthal and constant in magnitude at radius r. Choose a circular Amperian loop of radius r = 1.0 cm, coaxial with the wire. Since r < R, the loop is inside the conductor.
3
Step 3 — Compute I_enc

The current density is J = I / (πR²). The enclosed current through the loop of radius r is I_enc = J × πr² = I × (r²/R²) = 12 × (0.01² / 0.03²) = 12 × (1/9).

I_enc = 4/3 A ≈ 1.33 A
4
Step 4 — Apply Ampère's Law

∮ B · dl = B(2πr) = μ₀ I_enc. Solving for B: B = μ₀ I_enc / (2πr).
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Step 5 — Substitute and Solve

B = (4π × 10⁻⁷)(4/3) / (2π × 0.01) = (4π × 10⁻⁷ × 1.333) / (0.02π) = (5.333 × 10⁻⁷) / (0.02) = 2.67 × 10⁻⁵ T.

B ≈ 2.7 × 10⁻⁵ T = 27 μT
6
Step 6 — Check with Alternative Formula

We can verify using the direct formula B = μ₀Ir/(2πR²) = (4π × 10⁻⁷ × 12 × 0.01) / (2π × 0.0009) = (1.508 × 10⁻⁷) / (5.655 × 10⁻³) ≈ 2.67 × 10⁻⁵ T. ✓ Consistent.

SECTION 7

Ampère's Law vs. Biot–Savart Law

Students often wonder when to reach for Ampère's Law versus the Biot–Savart Law. The two are not competing tools; rather, they serve complementary roles, just as Gauss's law and Coulomb's law do in electrostatics. The choice depends entirely on the symmetry of the problem at hand.

Feature	Ampère's Law	Biot–Savart Law
Mathematical form	∮ B · dl = μ₀I_enc	dB = (μ₀/4π) I dl × r̂ / r²
When most useful	High-symmetry configurations (infinite wire, solenoid, toroid)	Arbitrary current geometries (finite wire, loops, arcs)
Gives B directly?	Yes, if B can be factored from the integral	Requires integration over the entire current distribution
Analogous electrostatics tool	Gauss's Law (∮ E · dA = Q/ε₀)	Coulomb's Law
Universality	Always true, but not always computationally useful	Always yields the field, though the integral may be difficult

When to use each law

SECTION 8

Connection to Maxwell's Equations

Ampère's original law works flawlessly for steady (DC) currents, but it breaks down when fields change with time. James Clerk Maxwell recognized in the 1860s that a time-varying electric flux through a surface produces the same magnetic effects as a real current. He introduced the displacement current term, ε₀ dΦ_E/dt, to patch the law. The corrected version—the Ampère–Maxwell law—is one of Maxwell's four equations and is essential for understanding electromagnetic waves.

Aspect	Ampère's Law (original)	Ampère–Maxwell Law
Equation	∮ B · dl = μ₀I_enc	∮ B · dl = μ₀I_enc + μ₀ε₀ dΦ_E/dt
Valid for	Steady (magnetostatic) currents only	All situations, including time-varying fields
Physical content	Real currents generate circulating B fields	Real currents AND changing electric flux generate circulating B fields
Consequence	Cannot explain EM wave propagation	Predicts electromagnetic waves traveling at c = 1/√(μ₀ε₀)

Original vs. generalized Ampère's Law

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AP Exam Note

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A circular Amperian loop encircles three parallel wires. Wire 1 carries 5 A into the page, wire 2 carries 3 A out of the page, and wire 3 carries 4 A into the page. A fourth wire carrying 10 A passes outside the loop. What is the value of ∮ B · dl around the loop (taking out of the page as positive)?

A. −6μ₀ T·m
B. −16μ₀ T·m
C. +6μ₀ T·m
D. −2μ₀ T·m

Show answer (A)

Only currents enclosed by the loop contribute to the line integral. Using the sign convention (out of page = positive): I_enc = −5 + 3 − 4 = −6 A. Therefore ∮ B · dl = μ₀(−6) = −6μ₀ ≈ −7.54 × 10⁻⁶ T·m. The fourth wire, despite carrying the largest current, is entirely outside the loop and contributes nothing to the circulation integral—though it does affect the local value of B at points on the loop.

Question 2: BASIC CALCULATION

A long straight wire carries a current of 8.0 A. What is the magnitude of the magnetic field at a perpendicular distance of 5.0 cm from the wire?

A. 16 μT
B. 32 μT
C. 64 μT
D. 25 μT

Show answer (B)

Apply B = μ₀I/(2πr) = (4π × 10⁻⁷ T·m/A)(8.0 A) / (2π × 0.050 m) = (4 × 10⁻⁷ × 8.0) / (0.10) = 3.2 × 10⁻⁶ / 0.10 = 3.2 × 10⁻⁵ T = 32 μT. This is a straightforward plug-in using the result derived from Ampère's Law for an infinite straight wire.

Question 3: INTERMEDIATE

A solenoid is 0.40 m long with 800 turns and carries a current of 3.0 A. What is the magnitude of the magnetic field at the center of the solenoid?

A. 1.9 mT
B. 3.0 mT
C. 7.5 mT
D. 15 mT

Show answer (C)

For an ideal solenoid, B = μ₀nI where n = N/L = 800/0.40 = 2000 turns/m. Thus B = (4π × 10⁻⁷)(2000)(3.0) = 4π × 10⁻⁷ × 6000 = 24π × 10⁻⁴ × 10⁻³ = 7.54 × 10⁻³ T ≈ 7.5 mT. This calculation requires correctly computing the turn density n first, then applying the solenoid formula derived via a rectangular Amperian loop.

Question 4: APPLIED

A coaxial cable consists of an inner solid conductor of radius a = 1.0 mm carrying current I = 5.0 A outward, surrounded by a thin outer cylindrical shell of radius b = 4.0 mm carrying the return current 5.0 A inward. Using Ampère's Law, determine the magnetic field at: (a) r = 3.0 mm (between the conductors), (b) r = 6.0 mm (outside both conductors). Justify each step.

Show answer

(a) At r = 3.0 mm, the Amperian loop encloses only the inner conductor's current: I_enc = 5.0 A. By symmetry, B(2πr) = μ₀I_enc, so B = μ₀(5.0)/(2π × 0.003) = (4π × 10⁻⁷ × 5.0)/(6π × 10⁻³) = (20 × 10⁻⁷)/(6 × 10⁻³) = 3.33 × 10⁻⁴ T ≈ 0.33 mT, directed circumferentially. (b) At r = 6.0 mm, the loop encloses both +5.0 A (inner) and −5.0 A (outer), so I_enc = 0. Therefore B = 0 outside the cable. This is why coaxial cables produce no external magnetic interference—a key engineering advantage.

Question 5: CRITICAL THINKING

A student claims: 'Since Ampère's Law says ∮ B · dl = μ₀I_enc, if I_enc = 0 for a given loop, then B = 0 everywhere on that loop.' Evaluate this claim. Provide a specific counterexample with a diagram or description, and explain the physical reasoning behind your answer.

Show answer

The student's claim is incorrect. Ampère's Law states that the integral of B · dl around the closed loop equals μ₀I_enc; this does not require B itself to be zero at each point on the loop—only that the positive and negative contributions to the line integral cancel. Counterexample: consider a circular Amperian loop centered midway between two parallel wires carrying equal currents in the same direction. The loop may enclose neither wire, giving I_enc = 0, yet B is clearly nonzero at every point on the loop because each wire produces a field that doesn't vanish there. The positive and negative dot-product contributions along different parts of the loop sum to zero, but the field magnitude at individual points is generally nonzero. This parallels the electrostatic case: Gauss's Law with Q_enc = 0 does not imply E = 0 on the Gaussian surface.