Ampère's Law - College Physics

SECTION 1

Historical Context & Motivation

The relationship between electricity and magnetism was one of the great scientific puzzles of the early nineteenth century. Before the 1820s, electric phenomena and magnetic phenomena were treated as entirely separate branches of physics; a compass needle and a Leyden jar seemed to belong to different worlds. The discovery that an electric current could deflect a compass needle shattered that partition and launched a feverish program of research that would ultimately culminate in Ampère's Law — a compact mathematical statement connecting the magnetic field circulating around a closed path to the total current threading through that path.

1820 • Ørsted's Discovery

Hans Christian Ørsted demonstrated that an electric current flowing through a wire deflects a nearby magnetic compass needle, providing the first evidence that electricity and magnetism are intimately linked.

1820 • Biot–Savart Law

Within months of Ørsted's announcement, Jean-Baptiste Biot and Félix Savart quantified the magnetic field produced by a small current element, establishing the differential form of the force law for magnetostatics.

1826 • Ampère's Circuital Law

André-Marie Ampère published a comprehensive mathematical framework showing that the line integral of the magnetic field around any closed loop equals μ₀ times the enclosed current. This integral relationship was far more practical than summing infinitesimal Biot–Savart contributions for symmetric geometries.

1865 • Maxwell's Extension

James Clerk Maxwell added the displacement-current term to Ampère's original law, correcting it for time-varying electric fields and completing the set of equations that unify electricity, magnetism, and optics.

Ampère's work addressed a critical question: given a known distribution of steady currents, how can we determine the magnetic field everywhere in space without performing a full vector integration over every current element? In situations possessing high symmetry — infinite straight wires, solenoids, toroids — Ampère's Law converts a difficult integral into a simple algebraic equation, making it one of the most powerful tools in the physicist's magnetostatics toolkit.

SECTION 2

Core Principles & Definitions

At its heart, Ampère's Law is a statement about the circulation of the magnetic field. Circulation is the line integral of a vector field around a closed path; when applied to B (the magnetic field), it measures how much the field wraps around a chosen loop. The law asserts that this circulation depends only on the net current passing through the loop, regardless of the loop's shape or size. Several foundational ideas make this statement precise and useful.

Amperian Loop

An imaginary closed curve chosen by the analyst to exploit the symmetry of the current distribution. The loop is not a physical object — it is a mathematical construct analogous to a Gaussian surface in electrostatics.

Enclosed Current (I_enc)

The total current piercing any surface bounded by the Amperian loop. A current is positive if it passes in the direction given by the right-hand rule relative to the loop's traversal direction; otherwise it is negative.

Line Integral of B

The quantity ∮ B · dl sums the component of the magnetic field tangent to the path over the entire loop. Only the tangential component contributes; perpendicular components integrate to zero.

Symmetry Requirement

Ampère's Law is always true, but it is most useful for calculating B only when the geometry lets you factor B out of the integral — e.g., cylindrical symmetry around a long straight wire, or translational symmetry inside a solenoid.

Permeability of Free Space (μ₀)

The constant μ₀ = 4π × 10⁻⁷ T·m/A sets the scale for magnetic effects in vacuum. It plays a role analogous to ε₀ in Gauss's law, connecting source (current) to field (B).

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KEY TAKEAWAY

Think of Ampère's Law like a toll-road analogy: imagine walking a circular path around a highway. The total 'toll' you accumulate (the line integral of B) depends only on how many cars (currents) pass through the area enclosed by your walking path — not on the specific shape of the path you chose, nor on how far from the road you walk. More cars through the loop means a bigger toll; currents outside the loop contribute nothing to the total.

SECTION 3

Visual Explanation — The Amperian Loop

A circular Amperian loop of radius r centered on a long, straight wire carrying current I out of the page. The magnetic field B is everywhere tangent to the loop and constant in magnitude by cylindrical symmetry. The infinitesimal path element dl is also tangent, so B · dl = B dl, and the integral reduces to B × 2πr = μ₀I.

The diagram above illustrates the canonical example of Ampère's Law: an infinitely long, straight wire. The current I emerges from the page (denoted by the ⊙ symbol), and the magnetic field forms concentric circles around the wire. Because of this perfect cylindrical symmetry, the magnitude of B is the same at every point on a circular loop of radius r. The direction of B is always tangent to the loop, which means the dot product B · dl simply becomes B dl. This allows us to pull B out of the integral and immediately solve for the field magnitude.

SECTION 4

Mathematical Framework

Ampère's Law has both an integral form, which is most useful for direct calculation in symmetric problems, and a differential form that connects naturally to Maxwell's equations. Below we present both, along with the key modifications introduced by Maxwell.

AMPÈRE'S LAW — INTEGRAL FORM

∮ B · dl = μ₀ I_enc

B = magnetic field (T); dl = infinitesimal path element along the Amperian loop (m); μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space); I_enc = net current enclosed by the loop (A). The circle on the integral sign (∮) indicates integration around a closed path.

MAGNETIC FIELD OF AN INFINITE STRAIGHT WIRE

B = μ₀ I / (2πr)

Derived by choosing a circular Amperian loop of radius r centered on the wire. By symmetry, B is constant and tangent to the loop, giving ∮ B · dl = B(2πr). Setting this equal to μ₀I and solving for B yields the formula above.

MAGNETIC FIELD INSIDE A SOLENOID

B = μ₀ n I

Here n = N/L is the number of turns per unit length, N is total turns, and L is the solenoid length. A rectangular Amperian loop with one side inside and one side outside shows that only the interior leg contributes to the integral; the field outside an ideal infinite solenoid is zero.

AMPÈRE–MAXWELL LAW (WITH DISPLACEMENT CURRENT)

∮ B · dl = μ₀ (I_enc + ε₀ dΦ_E/dt)

Maxwell's correction adds the displacement current term ε₀ dΦ_E/dt, where Φ_E is the electric flux through the surface bounded by the loop. This term ensures continuity of the magnetic field even when currents are interrupted, such as across the gap of a charging capacitor.

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Differential Form

In differential notation, Ampère's Law becomes ∇ × B = μ₀J, where J is the current density vector (A/m²). The curl of B at a point equals μ₀ times the current density at that point. This local form is equivalent to the integral form via Stokes' theorem.

SECTION 5

Applying Ampère's Law — Key Geometries

Ampère's Law is most powerful when applied to configurations possessing sufficient symmetry to simplify the line integral. The three classic geometries — the infinite straight wire, the ideal solenoid, and the toroid — each exploit a different type of symmetry (cylindrical, translational, and rotational, respectively). The table below summarizes the strategy and result for each.

Side-by-side comparison of the three classic Ampère's Law geometries. Each panel shows the Amperian loop (dashed cyan), the current-carrying conductors (violet), and the resulting B field direction (pink arrows). The derived formula and symmetry type are listed beneath each configuration.

Geometry	Amperian Loop Shape	Symmetry Exploited	Result for B
Infinite straight wire	Circle of radius r, centered on wire	Cylindrical — B constant on circle	B = μ₀I / (2πr)
Ideal solenoid	Rectangle with one side inside, one outside	Translational — B uniform inside, zero outside	B = μ₀nI
Toroid	Circle of radius r inside the torus	Rotational — B constant on circle inside winding	B = μ₀NI / (2πr)
Coaxial cable	Circle of radius r, coaxial with cable	Cylindrical — fields in different radial regions	Piecewise: depends on whether r is inside inner conductor, between conductors, or outside

Summary of standard Ampère's Law applications

The strategy for each geometry follows a common pattern: (1) identify the symmetry that makes B constant along portions of the loop, (2) choose an Amperian loop whose segments are either parallel or perpendicular to B, and (3) evaluate the integral ∮ B · dl by pulling B out of the integral where possible. For the solenoid, the rectangular loop is particularly elegant: two sides are perpendicular to B (contributing zero), one side is outside where B ≈ 0, and only the interior side of length l contributes, enclosing nI turns of current, so Bl = μ₀nIl and thus B = μ₀nI.

SECTION 6

Worked Example — Magnetic Field of a Coaxial Cable

A coaxial cable consists of an inner solid conductor of radius a = 2.0 mm carrying a current I = 5.0 A outward, surrounded by a thin outer cylindrical shell of radius b = 6.0 mm carrying a return current of 5.0 A inward. Determine the magnetic field at a distance r = 4.0 mm from the central axis (i.e., in the region between the two conductors).

Coaxial Cable — Field Between Conductors

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Step 1 — Identify the Region and Symmetry

The point of interest lies between the inner conductor (r = 2.0 mm) and the outer shell (r = 6.0 mm), so a < r < b. The cylindrical symmetry of the coaxial cable means the magnetic field is tangential and constant in magnitude on any circle concentric with the axis.
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Step 2 — Choose the Amperian Loop

Select a circular Amperian loop of radius r = 4.0 mm = 4.0 × 10⁻³ m, centered on the cable axis. By symmetry, B is tangent to this circle and has constant magnitude everywhere on it.
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Step 3 — Determine the Enclosed Current

The only conductor fully enclosed by the loop at r = 4.0 mm is the inner conductor carrying I = 5.0 A outward. The outer shell lies at r = 6.0 mm, outside our loop, and contributes nothing.

I_enc = 5.0 A
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Step 4 — Apply Ampère's Law

∮ B · dl = μ₀ I_enc. Since B is constant and tangent, the left side becomes B × (2πr). Therefore B × 2π(4.0 × 10⁻³) = (4π × 10⁻⁷)(5.0).
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Step 5 — Solve for B

B = μ₀I / (2πr) = (4π × 10⁻⁷ T·m/A)(5.0 A) / [2π(4.0 × 10⁻³ m)] = (20π × 10⁻⁷) / (8π × 10⁻³) = 2.5 × 10⁻⁴ T.

B = 2.5 × 10⁻⁴ T = 0.25 mT
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Step 6 — Verify Direction

Using the right-hand rule with the thumb pointing in the direction of the outward current (out of the page), the fingers curl counterclockwise. The magnetic field at r = 4.0 mm points counterclockwise (when viewed end-on with the current coming toward you).

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Note: Field Outside the Cable

At any radius r > b, the Amperian loop encloses both +5.0 A (inner) and −5.0 A (outer), giving I_enc = 0. Therefore B = 0 everywhere outside the cable — a key advantage of the coaxial design for minimizing electromagnetic interference.

SECTION 7

Strengths, Limitations & Comparison with Biot–Savart

Ampère's Law and the Biot–Savart Law are complementary approaches to magnetostatics. Both are rigorously correct for steady currents, but each has practical strengths in different contexts. The following comparison highlights when to reach for each tool.

Feature	Ampère's Law	Biot–Savart Law
Form	Integral (∮ B · dl = μ₀I_enc)	Integral (dB = μ₀/4π × Idl × r̂ / r²)
Best used when	High symmetry allows B to be factored out of the integral	Arbitrary current geometry; no special symmetry required
Computational ease	Simple algebra once the correct loop is identified	Requires vector integration over the entire current distribution
Information returned	Magnitude (and sometimes direction) of B along the loop	Full vector B at any specific point in space
Limitation	Useless for finding B if symmetry is insufficient to simplify the integral	Often analytically intractable; may require numerical methods
Analogy in electrostatics	Gauss's Law (∮ E · dA = Q_enc/ε₀)	Coulomb's Law (point-by-point integration)

Ampère's Law vs. Biot–Savart Law

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KEY TAKEAWAY

The relationship between Ampère's Law and the Biot–Savart Law mirrors the relationship between Gauss's Law and Coulomb's Law in electrostatics. Just as Gauss's Law provides a shortcut for calculating the electric field when the charge distribution is symmetric enough to choose a convenient Gaussian surface, Ampère's Law provides a shortcut for the magnetic field when the current distribution is symmetric enough to choose a convenient Amperian loop. When symmetry is lacking, you fall back to the more general (but harder) point-by-point integration.

SECTION 8

Connection to Maxwell's Equations & Displacement Current

Ampère's original law works flawlessly for steady (DC) currents, but it encounters a fundamental inconsistency when currents are time-varying. Consider a parallel-plate capacitor being charged by a current I. If you draw an Amperian loop around the wire leading to the capacitor, the enclosed current is I. But now inflate the surface bounded by that same loop so that it passes between the capacitor plates — no conduction current crosses that surface, implying I_enc = 0. The law gives two different answers depending on which surface you pick, which violates mathematical consistency.

Maxwell resolved this paradox by introducing the displacement current term I_d = ε₀ dΦ_E/dt. Between the capacitor plates, the electric field is changing as charge accumulates, so the electric flux through the surface is increasing. Maxwell showed that ε₀ dΦ_E/dt between the plates exactly equals the conduction current I in the wire, restoring consistency. The corrected law, ∮ B · dl = μ₀(I_enc + ε₀ dΦ_E/dt), is known as the Ampère–Maxwell Law and is one of the four Maxwell equations.

Aspect	Original Ampère's Law	Ampère–Maxwell Law
Equation	∮ B · dl = μ₀I_enc	∮ B · dl = μ₀(I_enc + ε₀ dΦ_E/dt)
Valid for	Steady (time-independent) currents only	All situations, including time-varying fields
Predicts EM waves?	No	Yes — combined with Faraday's Law, it yields the wave equation for light
Capacitor paradox	Gives inconsistent results across different surfaces	Fully consistent: displacement current fills the gap

Original vs. corrected Ampère's Law

The displacement current was the final ingredient that allowed Maxwell to predict the existence of electromagnetic waves propagating at the speed of light. In your future study of optics and wave theory, the Ampère–Maxwell Law will serve as one of the cornerstones showing that light is an electromagnetic phenomenon. For the purposes of magnetostatics — steady currents with no time-varying fields — the displacement current term vanishes, and you can safely use Ampère's original formulation.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A long straight wire carries a steady current I. An Amperian loop in the shape of a square (rather than a circle) is drawn around the wire, with the wire passing through the center of the square. Does the value of ∮ B · dl depend on whether the loop is circular or square? Explain.

Show answer

No. Ampère's Law states that ∮ B · dl = μ₀I_enc, and the enclosed current I_enc is the same regardless of the loop shape. The value of the line integral around the square is still μ₀I, exactly the same as for a circular loop. However, the circular loop is preferred for calculation purposes because the cylindrical symmetry makes B constant in magnitude and tangent to the loop everywhere, allowing B to be factored out of the integral. On a square loop, B varies in both magnitude and angle along the path, so while the integral still equals μ₀I, you cannot easily extract B from it.

Question 2: BASIC CALCULATION

A long, straight wire carries a current of 12.0 A. Calculate the magnitude of the magnetic field at a perpendicular distance of 8.0 cm from the wire. (μ₀ = 4π × 10⁻⁷ T·m/A)

Show answer

Using B = μ₀I / (2πr), we substitute I = 12.0 A and r = 0.080 m. B = (4π × 10⁻⁷)(12.0) / [2π(0.080)] = (48π × 10⁻⁷) / (0.160π) = (48 × 10⁻⁷) / 0.160 = 3.0 × 10⁻⁵ T = 30 μT. This is comparable to Earth's magnetic field in many locations, illustrating why household wires carrying modest currents can influence nearby compasses.

Question 3: INTERMEDIATE

A solenoid is 0.50 m long, has 800 turns, and carries a current of 3.0 A. (a) Calculate the magnetic field inside the solenoid. (b) If the solenoid has a circular cross-section with radius 2.0 cm, what is the total magnetic flux through one turn?

Show answer

(a) The turn density is n = N/L = 800/0.50 = 1600 turns/m. Applying Ampère's Law for a solenoid: B = μ₀nI = (4π × 10⁻⁷)(1600)(3.0) = 6.03 × 10⁻³ T ≈ 6.0 mT. (b) The magnetic flux through one turn is Φ = BA = B × πr² = (6.03 × 10⁻³)(π)(0.020)² = (6.03 × 10⁻³)(1.257 × 10⁻³) = 7.6 × 10⁻⁶ Wb ≈ 7.6 μWb. The field inside is uniform (by the solenoid approximation), so we simply multiply B by the cross-sectional area.

Question 4: APPLIED

A toroidal coil has 500 turns wound uniformly on a doughnut-shaped core with an inner radius of 10.0 cm and an outer radius of 14.0 cm. The coil carries a current of 2.0 A. Determine the magnetic field at (a) the midpoint of the winding (r = 12.0 cm) and (b) a point outside the toroid (r = 20.0 cm).

Show answer

(a) For a toroid, B = μ₀NI / (2πr). At r = 0.12 m: B = (4π × 10⁻⁷)(500)(2.0) / [2π(0.12)] = (4000π × 10⁻⁷) / (0.24π) = (4000 × 10⁻⁷) / 0.24 = 1.67 × 10⁻³ T ≈ 1.7 mT. (b) At r = 0.20 m, the Amperian loop lies entirely outside the toroid. By Ampère's Law, the loop encloses both the outgoing and returning portions of each turn, so the net enclosed current is zero. Therefore B = 0 at r = 20.0 cm. This confinement of the field to the interior of the winding is a distinctive and useful property of toroids, exploited in transformer cores and tokamak fusion reactors.

Question 5: CRITICAL THINKING

A thick cylindrical conductor of radius R carries a total current I uniformly distributed over its cross-section. Derive an expression for the magnetic field B as a function of radial distance r for both r < R and r > R. Show that B is continuous at r = R, and sketch B(r).

Show answer

For r > R: The entire current I is enclosed, so by Ampère's Law ∮ B · dl = B(2πr) = μ₀I, giving B = μ₀I / (2πr). This falls off as 1/r, identical to the thin-wire result. For r < R: The current is uniformly distributed, so the current density is J = I / (πR²). A circular Amperian loop of radius r encloses I_enc = J × πr² = I(r²/R²). Then B(2πr) = μ₀I(r²/R²), so B = μ₀Ir / (2πR²). This grows linearly with r inside the conductor. At r = R, the interior formula gives B = μ₀I(R) / (2πR²) = μ₀I / (2πR), which exactly matches the exterior formula evaluated at r = R, confirming continuity. The sketch shows B rising linearly from 0 at r = 0 to a maximum of μ₀I / (2πR) at r = R, then decreasing as 1/r for r > R.

SUMMARY

Summary — Ampère's Law

Summary

Ampère's Law establishes that the line integral of the magnetic field around any closed Amperian loop equals μ₀ times the net enclosed current. When sufficient symmetry exists — cylindrical for an infinite straight wire, translational for a solenoid, or rotational for a toroid — the magnetic field can be extracted from the integral with simple algebra, yielding classic results such as B = μ₀I/(2πr) and B = μ₀nI.

The law is analogous to Gauss's Law in electrostatics and stands as one of Maxwell's four equations when extended with the displacement current term ε₀ dΦ_E/dt. This correction, due to Maxwell, reconciles the law with time-varying fields and ultimately predicts electromagnetic waves. For magnetostatics problems, the original form remains the go-to tool whenever symmetry permits.