Ampère’s Law
Ampère’s law relates the circulation of the magnetic field around a closed path to the net current threading that path. In AP Physics C: E&M, it becomes a powerful field-solving tool only when symmetry makes B⃗·dℓ⃗ simple—especially for long straight wires, cylindrical conductors, current-carrying slabs, and long solenoids. The full Ampère–Maxwell law also explains conceptually that a changing electric field can create a magnetic field.
Topic sequence · 4 of 4
Learning objectives
Objective 1
Choose and justify an Amperian loop, evaluate ∮B⃗·dℓ⃗ with correct sign and geometry, and distinguish the always-valid law from the narrower set of symmetric situations in which it directly solves for B.
Objective 2
Derive magnetic-field expressions for long straight wires, cylindrical current distributions, and long solenoids, including cases where the enclosed current must be obtained from I_enc=∫J⃗·dA⃗.
Objective 3
Connect Ampère’s law to Biot–Savart, superposition, and Maxwell’s equations while respecting the course boundary: changing electric flux should be understood as a magnetic-field source, but the displacement-current term is not a required quantitative calculation here.
- Right-hand-rule direction for magnetic fields created by current from Topics 12.2–12.3.
- Line integrals and dot products: B⃗·dℓ⃗=B dℓ cosθ, including the meaning of a closed-path integral.
- Current density from Unit 11, including I=∫J⃗·dA⃗ for a nonuniform current distribution.
- Symmetry reasoning from Gauss’s law: a law can be universally true while only certain geometries make it an efficient field-solving shortcut.
Definitions first
Representations you must read and create
Representation 1
Circular Amperian loop around a long straight wire: draw B⃗ tangent everywhere, dℓ⃗ tangent in the chosen traversal direction, and mark the enclosed current with ⊙ or ⊗ notation before assigning signs.
Representation 2
B-versus-r graph for a uniformly current-filled cylindrical wire: B grows linearly from zero inside, reaches μ₀I/(2πR) at the surface, then decreases as 1/r outside. The graph must be continuous at r=R.
Representation 3
Current-density cross section: shade only the area inside radius r, write dA=2πs ds for concentric annuli, and integrate J(s) from s=0 to s=r when J is nonuniform.
Representation 4
Long-solenoid Amperian rectangle: one long side lies inside parallel to B, the opposite long side lies outside where the ideal-model field is negligible, and the short sides contribute zero because B⃗⊥dℓ⃗.
Representation 5
Superposition diagram: calculate fields from each conductor separately with direction, then add vectors at the observation point. Ampère’s law does not mean outside currents have zero local field; it means their net contribution to the closed circulation can cancel.
Representation 6
Charging-capacitor conceptual diagram: the same loop can bound a surface cutting a wire or bulging between capacitor plates. The Maxwell addition makes the magnetic-field law independent of which spanning surface is imagined.
Core relationships and calculus
Variables: B T; dℓ m; μ₀=4π×10⁻⁷ T·m/A; I_enc A
Use: Use for steady-current magnetic fields. Traverse a closed path with an orientation; current sign follows the corresponding right-hand-rule surface normal.
Variables: I_enc A; J A/m²; dA m²
Use: Use when current density is distributed across a conductor. Integrate only over the portion of the conductor enclosed by the Amperian loop.
Variables: B T; I A; r m
Use: Field magnitude outside a very long straight wire. The field is tangent to circles centered on the wire.
Variables: B T; I A; r,R m
Use: Inside a solid cylindrical wire of radius R carrying uniformly distributed total current I. The enclosed-current fraction is r²/R².
Variables: B T; n turns/m; I A
Use: Approximate field inside a long air-core solenoid, with n=N/ℓ. Outside-field magnitude is negligible in the ideal long-solenoid model.
Variables: B T; I A; Φ_E N·m²/C; ε₀ F/m
Use: Full Ampère–Maxwell equation. In this course, understand qualitatively that changing electric flux creates magnetic field; do not treat the displacement-current term as a required quantitative Ampère-law calculation.
The calculus is a closed line integral paired with an area integral for the source. For a symmetric circular path around a long straight current, B⃗ is parallel to dℓ⃗ and has constant magnitude, so ∮B⃗·dℓ⃗=B∮dℓ=B(2πr). If the current density is nonuniform, however, the right side is not automatically μ₀I_total. Write I_enc=∫J⃗·dA⃗. In cylindrical symmetry, dA=2πs ds, so I_enc(r)=∫₀ʳJ(s)2πs ds. This two-integral structure—one around the field path, one through the source area—is the central Physics C reasoning pattern.
For an ideal long solenoid, use a rectangular Amperian loop of length L with one long segment inside and parallel to the approximately uniform axial field. The outside long segment contributes approximately zero; the two short segments contribute zero because B⃗·dℓ⃗=0 there. If the solenoid has n turns per unit length, then nL turns pierce the spanning surface, each carrying current I, so I_enc=nLI. Ampère’s law becomes BL=μ₀nLI, hence B=μ₀nI. The cancellation of L confirms that the ideal interior field depends on turns per length rather than the arbitrary loop length.
Use quantitative Ampère-law methods only where magnetic symmetry supports them: long straight wires, long solenoids, conductive slabs, and cylindrical conductors/current-density distributions are the core shapes. Do not assume an arbitrary wire bend, finite solenoid end region, or asymmetric current network permits B to be pulled outside the line integral. The full Ampère–Maxwell equation is conceptually important because changing electric fields create magnetic fields, but the displacement-current term is not a required quantitative calculation in this topic. Toroids are a useful college-physics extension but should not displace the explicitly required geometries.
Physical examples
Example 1
A 5.0 A current in a very long straight wire creates circular magnetic field lines. At r=4.0 cm, symmetry makes B constant and tangent along a circular loop, so B(2πr)=μ₀I.
Example 2
Inside a thick wire with uniform current density, an Amperian loop at half the wire radius encloses only one-quarter of the total current, not one-half, because enclosed current scales with area.
Example 3
If current density increases toward the center or edge, do not use I_enc=I(r²/R²). Integrate J(r) over the enclosed cross-sectional area first, then apply Ampère’s law.
Example 4
For an ideal long solenoid with n=1200 turns/m and I=0.80 A, the interior field is approximately μ₀nI. Reversing the current reverses the field direction without changing its magnitude.
Example 5
For two long parallel wires, the net field at a point is a vector sum. An Amperian loop enclosing both currents gives information about total circulation, but a separate symmetry argument is still needed to infer the field at a particular point.
Example 6
A loop around a charging-capacitor circuit has a nonzero magnetic circulation even if a chosen spanning surface passes between the plates where no conduction charge crosses. A changing electric flux supplies the missing source term in the complete Maxwell equation.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Before integrating, state why B has constant magnitude on the chosen path and why it is parallel, antiparallel, or perpendicular to dℓ⃗ on each segment. The loop choice is part of the solution, not decoration.
Reasoning pattern 2
Compute I_enc from geometry or J(r). For a thick wire, the current enclosed changes with loop radius even though the total wire current is fixed.
Reasoning pattern 3
Distinguish enclosed current from nearby current. A current outside the loop can make B nonzero at every point on the path while contributing zero net amount to ∮B⃗·dℓ⃗.
Reasoning pattern 4
Check continuity at a conductor boundary. For a finite, nonsingular volume current density, an inside and outside expression should normally meet at the surface.
Reasoning pattern 5
Use right-hand-rule orientation consistently: reversing the integration direction reverses the positive surface normal and therefore reverses the sign assigned to I_enc.
Reasoning pattern 6
When the geometry lacks sufficient symmetry, say so and use Biot–Savart or superposition instead of forcing Ampère’s law into B×path-length form.
Deeper calculus and limiting-case check
Let a cylindrical wire of radius R carry J(s)=J₀(1−s²/R²). Then I_enc(r)=2πJ₀∫₀ʳ(s−s³/R²)ds=πJ₀[r²−r⁴/(2R²)]. The total current is I=πJ₀R²/2, so for r<R, I_enc=I[2r²/R²−r⁴/R⁴]. Ampère’s law gives B(r)=μ₀I/(2πr)[2r²/R²−r⁴/R⁴]. At r→0 the field approaches zero linearly; at r=R the bracket becomes 1 and the expression matches the exterior field μ₀I/(2πR). This limiting-case and boundary check is a strong defense against algebra errors.
Model-selection workflow: decide what the problem is before choosing an equation
For Ampère’s Law, the fastest reliable route is not equation hunting. Start by identifying the current/charge source and field geometry, encode it with the magnetic vector representation, choose the Lorentz/Biot–Savart/Ampere equation, and then test the cross-product and symmetry assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat Ampère’s law and Amperian loop as physical quantities first and symbols second.
Choose a representation
Build Circular Amperian loop around a long straight wire: draw B⃗ tangent everywhere, dℓ⃗ tangent in the chosen traversal direction, and mark the enclosed current with ⊙ or ⊗ notation before assigning signs.. If the prompt supplies B-versus-r graph for a uniformly current-filled cylindrical wire: B grows linearly from zero inside, reaches μ₀I/(2πR) at the surface, then decreases as 1/r outside. The graph must be continuous at r=R., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is ∮ B⃗·dℓ⃗ = μ₀ I_enc. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
Use quantitative Ampère-law methods only where magnetic symmetry supports them: long straight wires, long solenoids, conductive slabs, and cylindrical conductors/current-density distributions are the core shapes. Do not assume an arbitrary wire bend, finite solenoid end region, or asymmetric current network permits B to be pulled outside the line integral. The full Ampère–Maxwell equation is conceptually important because changing electric fields create magnetic fields, but the displacement-current term is not a required quantitative calculation in this topic. Toroids are a useful college-physics extension but should not displace the explicitly required geometries. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Circular Amperian loop around a long straight wire: draw B⃗ tangent everywhere, dℓ⃗ tangent in the chosen traversal direction, and mark the enclosed current with ⊙ or ⊗ notation before assigning signs. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | ∮ B⃗·dℓ⃗ = μ₀ I_enc | Use for steady-current magnetic fields. Traverse a closed path with an orientation; current sign follows the corresponding right-hand-rule surface normal.. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | I_enc = ∫_S J⃗·dA⃗ | Use when current density is distributed across a conductor. Integrate only over the portion of the conductor enclosed by the Amperian loop.. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | The calculus is a closed line integral paired with an area integral for the source. For a symmetric circular path around a long straight current, B⃗ is parallel to dℓ⃗ and has constant magnitude, so ∮B⃗·dℓ⃗=B∮dℓ=B(2πr). If the current density is nonuniform, however, the right side is not automatically μ₀I_total. Write I_enc=∫J⃗·dA⃗. In cylindrical symmetry, dA=2πs ds, so I_enc(r)=∫₀ʳJ(s)2πs ds. This two-integral structure—one around the field path, one through the source area—is the central Physics C reasoning pattern. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A solid cylindrical wire of radius R=6.00 mm carries a uniformly distributed current I=12.0 A out of the page. Find B at r=3.00 mm and at r=12.0 mm. For r<R, choose a circular Amperian loop. Uniform current density means I_enc=I(r²/R²)=12.0(3.00²/6.00²)=3.00 A. Because B is tangent and constant on the loop, B(2πr)=μ₀I_enc, so B=(4π×10⁻⁷)(3.00)/[2π(0.00300)]=2.00×10⁻⁴ T. For r>R, the loop encloses all 12.0 A, giving B=(4π×10⁻⁷)(12.0)/[2π(0.0120)]=2.00×10⁻⁴ T. The equal values are not a coincidence for these chosen radii. The interior linear law gives B∝r, while the exterior law gives B∝1/r; both meet continuously at r=R, where B=μ₀I/(2πR)=4.00×10⁻⁴ T. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 12.4
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship ∮ B⃗·dℓ⃗ = μ₀ I_enc should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Circular Amperian loop around a long straight wire: draw B⃗ tangent everywhere, dℓ⃗ tangent in the chosen traversal direction, and mark the enclosed current with ⊙ or ⊗ notation before assigning signs. and B-versus-r graph for a uniformly current-filled cylindrical wire: B grows linearly from zero inside, reaches μ₀I/(2πR) at the surface, then decreases as 1/r outside. The graph must be continuous at r=R.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Run a magnetic-field mapping investigation with a long straight conductor or solenoid and a Hall sensor/phone magnetometer whose limitations are documented. For the straight conductor, record B at multiple radial distances for at least two currents, subtract background field by reversing current, and graph B versus 1/r outside the wire. For a solenoid, measure B along the axis and compare the central region with μ₀nI while identifying end effects as departures from the ideal long-solenoid model. Include sensor zeroing, distance uncertainty, current stability, repeated trials, and a claim supported by the graph rather than by one data point.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Writing ∮B⃗·dℓ⃗=B(2πr) before establishing that B is constant and tangent along that path. | Return to the magnetic vector representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using total current when the Amperian loop lies inside a thick conductor and encloses only part of the current. | Return to the Lorentz/Biot–Savart/Ampere equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using I_enc=I(r/R) instead of an area-based or integrated current fraction. | Return to the magnetic vector representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming currents outside the loop create no magnetic field. They can create local field even though they do not change the net enclosed-current term. | Return to the Lorentz/Biot–Savart/Ampere equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Confusing the closed line integral in Ampère’s law with the closed surface integral in Gauss’s law for magnetism. | Return to the magnetic vector representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Applying B=μ₀nI near a short solenoid’s end without acknowledging that the ideal long-solenoid approximation is breaking down. | Return to the Lorentz/Biot–Savart/Ampere equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Quantitatively applying the Maxwell displacement-current term beyond the stated course boundary instead of keeping it conceptual. | Return to the magnetic vector representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 12.3 derives magnetic fields by summing current elements with Biot–Savart. Topic 12.4 reaches selected results faster by exploiting symmetry in a closed line integral. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Unit 11 current density supplies I_enc=∫J⃗·dA⃗, which is essential for thick-wire Ampère problems with nonuniform current density. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Gauss’s law in Unit 8 and Ampère’s law share a strategy: choose a mathematical boundary matched to symmetry, simplify an integral, and relate a field to an enclosed source. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write ∮ B⃗·dℓ⃗ = μ₀ I_enc and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 12.4 mastery ladder
Interactive Ampère-law field analyzer
Compare the magnetic field of a uniformly current-filled cylindrical wire at radius r with the ideal long-solenoid field. Inside the wire the tool uses the enclosed-current fraction r²/R²; outside it uses the full current.
Experimental connection
Run a magnetic-field mapping investigation with a long straight conductor or solenoid and a Hall sensor/phone magnetometer whose limitations are documented. For the straight conductor, record B at multiple radial distances for at least two currents, subtract background field by reversing current, and graph B versus 1/r outside the wire. For a solenoid, measure B along the axis and compare the central region with μ₀nI while identifying end effects as departures from the ideal long-solenoid model. Include sensor zeroing, distance uncertainty, current stability, repeated trials, and a claim supported by the graph rather than by one data point.
Connections across Physics C
Connection 1
Topic 12.3 derives magnetic fields by summing current elements with Biot–Savart. Topic 12.4 reaches selected results faster by exploiting symmetry in a closed line integral.
Connection 2
Unit 11 current density supplies I_enc=∫J⃗·dA⃗, which is essential for thick-wire Ampère problems with nonuniform current density.
Connection 3
Gauss’s law in Unit 8 and Ampère’s law share a strategy: choose a mathematical boundary matched to symmetry, simplify an integral, and relate a field to an enclosed source.
Connection 4
Topic 13.1 introduces magnetic flux Φ_B; Topic 13.2 then shows that changing magnetic flux creates an electric effect, complementing the Ampère–Maxwell statement that changing electric flux creates a magnetic effect.
Connection 5
The four Maxwell equations unify electric charge, magnetic-field closure, electromagnetic induction, current, and changing fields into one classical field framework.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Writing ∮B⃗·dℓ⃗=B(2πr) before establishing that B is constant and tangent along that path.
Avoid this
Using total current when the Amperian loop lies inside a thick conductor and encloses only part of the current.
Avoid this
Using I_enc=I(r/R) instead of an area-based or integrated current fraction.
Avoid this
Assuming currents outside the loop create no magnetic field. They can create local field even though they do not change the net enclosed-current term.
Avoid this
Confusing the closed line integral in Ampère’s law with the closed surface integral in Gauss’s law for magnetism.
Avoid this
Applying B=μ₀nI near a short solenoid’s end without acknowledging that the ideal long-solenoid approximation is breaking down.
Avoid this
Quantitatively applying the Maxwell displacement-current term beyond the stated course boundary instead of keeping it conceptual.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
What does Ampère’s law calculate?
It relates magnetic-field circulation around a closed path to net enclosed current. With strong symmetry it can also be rearranged to obtain B directly.
Is Ampère’s law always valid?
The complete Maxwell form is fundamental. The steady-current form is valid for magnetostatic situations, but direct field calculation is simple only when symmetry constrains B along the chosen path.
What current is enclosed by an Amperian loop?
The algebraic current crossing any surface whose boundary is that loop. For a distributed current, calculate it by integrating J⃗·dA⃗ over the enclosed portion.
Why does B grow linearly inside a uniform cylindrical wire?
Uniform J makes I_enc proportional to r², while the circular path length is proportional to r, so B∝I_enc/r∝r.
Why is the magnetic field inside a long solenoid nearly uniform?
For the ideal long-solenoid model, symmetry and the dense winding make the interior axial field nearly constant away from the ends while exterior contributions largely cancel.
What did Maxwell add to Ampère’s law?
A changing electric-flux term. It explains that changing electric fields can create magnetic fields and resolves the charging-capacitor consistency problem.