---
title: "Enclosed Current — AP Physics C: E&M Definition & Guide"
description: "Enclosed current is the net current passing through an Amperian loop, the I in Ampère's law (∮B·dl = μ₀I_enc). Getting it right is the whole game in Topic 12.4."
canonical: "https://fiveable.me/ap-physics-c-e-m/key-terms/enclosed-current"
type: "key-term"
subject: "AP Physics C: E&M"
unit: "Unit 12"
---

# Enclosed Current — AP Physics C: E&M Definition & Guide

## Definition

Enclosed current (I_enc) is the net electric current passing through the surface bounded by an Amperian loop. In Ampère's law, ∮B⃗·dl⃗ = μ₀I_enc, so only the current threading the loop determines the line integral of the magnetic field, not the total current in the conductor.

## What It Is

Enclosed current is the net [current](/ap-physics-c-e-m/unit-11/4-electric-power/study-guide/u2cRqQTlthIAJtwp "fv-autolink") that actually pokes through the imaginary surface bounded by your [Amperian loop](/ap-physics-c-e-m/key-terms/amperian-loop "fv-autolink"). Ampère's law says ∮B⃗·dl⃗ = μ₀I_enc, and the right side only counts current inside the loop. Current outside the loop still creates a magnetic field everywhere, but its contribution to the line integral around the loop cancels out perfectly. That's the subtle (and very testable) part.

Finding I_enc is usually the hardest step of an Ampère's law problem. If the loop encloses the whole wire, I_enc is just I. If you're inside a [conductor](/ap-physics-c-e-m/unit-11/3-resistance-resistivity-and-ohms-law/study-guide/TnRPkql9C75GQe0d "fv-autolink"), you only count the fraction of current inside radius r. For uniform current density that's I_enc = I(r²/R²), and for non-uniform density like J = kr, you have to integrate, I_enc = ∫J dA = ∫₀ʳ (kr')(2πr' dr'). Currents in opposite directions partially cancel, so 'net' really means net. Two equal and opposite currents through the same loop give I_enc = 0.

## Why It Matters

Enclosed current lives in Topic 12.4 (Ampère's Law) in [AP Physics C: E&M](/ap-physics-c-e-m "fv-autolink"), and it's the variable that makes or breaks every Ampère's law calculation. The [symmetry](/ap-physics-c-e-m/key-terms/symmetry "fv-autolink") argument tells you ∮B⃗·dl⃗ = B(2πr) for a circular loop, but B itself depends entirely on what you decide I_enc is. Coaxial cables, hollow cylinders, slabs of current, and solenoids all reduce to the same question. Draw a loop, figure out what current threads it, solve for B. If you can set up I_enc correctly, the rest is algebra. It's also conceptually deep, since it explains why the field inside a hollow current-carrying cylinder is zero (no enclosed current means the integral vanishes) even though there's plenty of current nearby.

## Connections

### Ampère's Law (Unit 12)

Enclosed current is the right-hand side of [Ampère's law](/ap-physics-c-e-m/unit-12/4-amperes-law/study-guide/RURo66Hv1aueyDWX "fv-autolink"), the way enclosed charge is the right-hand side of Gauss's law. The whole strategy of Topic 12.4 is choosing a loop where symmetry makes the integral easy, then carefully counting I_enc.

### [Magnetic field of a solenoid (Unit 12)](/ap-physics-c-e-m/key-terms/magnetic-field-of-a-solenoid)

The famous B = μ₀nI result comes straight from counting enclosed current. A rectangular Amperian loop that threads N turns of the [solenoid](/ap-physics-c-e-m/key-terms/solenoid "fv-autolink") encloses I_enc = NI, and that's where the turns-per-length factor n comes from.

### Gauss's Law and enclosed charge (Unit 8)

Enclosed current is the magnetic cousin of enclosed charge. In Gauss's law, only charge inside the surface sets the flux; in Ampère's law, only current through the loop sets the [line integral](/ap-physics-c-e-m/key-terms/line-integral "fv-autolink"). If you mastered q_enc with non-uniform charge density, the integral for I_enc with J = kr is the exact same move.

### [Permeability (Unit 12)](/ap-physics-c-e-m/key-terms/permeability)

The constant μ₀ is what converts enclosed current into the line integral of B. It plays the same role for magnetism that ε₀ (sitting in the denominator) plays for Gauss's law in electrostatics.

## On the AP Exam

Ampère's law problems almost always hinge on computing I_enc, and the exam loves making that step nontrivial. Multiple-choice stems give you a loop around a straight wire and ask for ∮B⃗·dl⃗ (answer: μ₀I, regardless of the loop's shape, as long as the wire passes through it). Harder questions hand you a non-uniform current density like J = kr and ask for B at r = R/2, which forces you to integrate J over the enclosed area instead of grabbing the total I. Hollow cylinders test whether you know I_enc = 0 for r < a, so B = 0 there. Current-carrying slabs ask how B varies with distance from the central plane, where I_enc grows linearly with z. On FRQs, expect to draw or describe your Amperian loop, justify the symmetry, write Ampère's law, and explicitly set up I_enc before solving for B. Skipping the I_enc setup is where points die.

## enclosed current vs Total current

The total current I is everything flowing in the conductor; the enclosed current I_enc is only the part passing through your Amperian loop. They're equal when your loop wraps around the entire conductor, but inside a wire of radius R at distance r < R, only a fraction is enclosed. For uniform current density, I_enc = I(r²/R²). Plugging total I into Ampère's law inside a conductor is one of the most common wrong answers on these problems.

## Key Takeaways

- Enclosed current is the net current passing through the surface bounded by an Amperian loop, and it equals ∮B⃗·dl⃗ divided by μ₀.
- Only current threading the loop counts; currents outside the loop affect B at individual points but contribute zero to the line integral around the loop.
- Inside a conductor with uniform current density, I_enc = I(r²/R²), so the field grows linearly with r inside and falls off as 1/r outside.
- For non-uniform current density like J = kr, find I_enc by integrating J over the enclosed area using rings: I_enc = ∫₀ʳ J(r')(2πr')dr'.
- Inside the hollow region of a cylindrical conductor (r < inner radius), I_enc = 0, so the magnetic field there is zero.
- Currents in opposite directions through the same loop subtract, which is exactly why the field outside an ideal coaxial cable is zero.

## FAQs

### What is enclosed current in Ampère's law?

It's the net electric current passing through the surface bounded by an Amperian loop. Ampère's law states ∮B⃗·dl⃗ = μ₀I_enc, so this enclosed current is what determines the line integral of the magnetic field around the loop.

### Does current outside the Amperian loop affect the magnetic field?

Yes and no, and this trips people up. Outside current does contribute to B at every point on the loop, but its contributions to the line integral ∮B⃗·dl⃗ cancel out exactly, so it adds nothing to μ₀I_enc. That's why Ampère's law only works cleanly when symmetry lets you pull B out of the integral.

### How is enclosed current different from total current?

Total current is everything the conductor carries; enclosed current is only what passes through your loop. Inside a uniform wire at r < R, I_enc = I(r²/R²), not I. They only match when the loop encloses the whole conductor.

### How do you find enclosed current with a non-uniform current density?

Integrate the current density over the enclosed area. For J = kr in a cylindrical wire, slice the cross-section into rings of area 2πr'dr', so I_enc = ∫₀ʳ kr'(2πr')dr' = 2πkr³/3. This is the same technique as finding enclosed charge for a non-uniform charge density in Gauss's law.

### Is the magnetic field zero inside a hollow current-carrying cylinder?

Yes, for points inside the hollow region (r less than the inner radius). An Amperian loop there encloses zero current, and by symmetry B(2πr) = μ₀(0), so B = 0 even though current flows in the surrounding shell.

## Related Study Guides

- [12.4 Ampère's Law](/ap-physics-c-e-m/unit-12/4-amperes-law/study-guide/RURo66Hv1aueyDWX)

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