UY1: Applications Of Ampere's Law

This page gives the UY1 working model/result for Applications Of Ampere’s Law. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

Module path: Electromagnetism (UY1)
Practice: UY1 Electromagnetism Quiz
Full routing: UY1 Assessment Map
Math toolkit: Mathematics for Undergraduate Physics

1) At a glance

These are magnetostatic field models (steady currents, no time-varying electric-flux term). For time-varying situations, see Displacement Current and Faraday’s Law Of Induction & Lenz’s Law.
Long solenoid (long, tightly wound; away from the ends):

Bᵢnside ≈ μ₀ nI, n = N/ell

Toroid (closely wound turns; field mostly confined to the core region):

B(r) = μ₀NI/2π r (inside the core region)

Infinite sheet current K (surface current density, A m⁻¹):

B = μ₀K/2quadon each side (opposite directions)

Quick checks: solenoid field is approximately uniform inside; toroid field falls like 1/r; sheet-current field is approximately constant with distance.

Prerequisites: Ampere’s Law, Magnetic Field Of A Straight Current Carrying Conductor, Magnetic Field Of A Circular Current Loop
Next uses: Electromagnetic Induction Experiments, Self-Inductance & Inductors

2) Setup

Apply

ointvecB · dvecell = μ₀Iₑnc

with loops aligned to each geometry.

Assumptions used:

Long solenoid: length much larger than radius, dense winding.
Toroid: closely wound turns around ring.
Current sheet: effectively infinite extent.
Steady currents (magnetostatics) and negligible fringing/edge fields in the regions where the formulas are applied.

Mixing N (total turns) with n (turns per unit length): n = N/ell.
Using the solenoid formula right at the ends: the outside field is not exactly zero and the inside field is not perfectly uniform near edges.
For toroids: using one radius when the field point is at another radius. The model is explicitly B(r)propto 1/r.
For a sheet current: K is A m⁻¹ (current per unit width), not A m⁻².

3) Core derivation/explanation

Long solenoid

Take a rectangular Amperian loop partly inside, partly outside.

Outside field is approximately zero.
Inside field is approximately uniform and parallel to loop segment.
Enclosed current for length ell is (nell)I.

So:

Bell ≈ μ₀(nell)I Rightarrow B ≈ μ₀nI.

Toroid

Use circular loop of radius r concentric with toroid.

vecB tangential and approximately constant on loop.
Enclosed current is NI if loop is in core region.

Thus:

B(2π r) = μ₀NI Rightarrow B(r) = μ₀NI/2π r.

Infinite current sheet

Using a rectangular loop straddling sheet with surface current density K:

2Bell = μ₀Kell Rightarrow B = μ₀K/2

with opposite directions on opposite sides.

Checks (sanity)

Units: μ₀ n I and μ₀ N I/(2π r) both give tesla.
Solenoid: if n → 0 (no turns per length), B → 0.
Toroid: larger r gives smaller field; outside the windings the ideal model predicts very small field (stronger confinement than a straight wire).
Sheet: the magnitude does not depend on distance from the sheet (ideal infinite-extent model).

4) Worked example(s)

Example 1: solenoid

A long solenoid has turn density n = 1500 m⁻¹ and current I = 0.30 A.

B = μ₀nI = (4π × 10⁻⁷)(1500)(0.30) = 5.65 × 10⁻⁴ T.

Example 2: toroid

A toroid has N = 400, I = 0.50 A, and point of interest at r = 8.0 cm.

B = μ₀NI/2π r = (4π × 10⁻⁷)(400)(0.50)/2π(8.0 × 10⁻²) = 5.0 × 10⁻⁴ T.

5) Practice set (with hints + answers)

A long solenoid has n = 2000 m⁻¹, I = 0.25 A. Find inside field.
For a toroid, if r doubles (same N,I), what happens to B?
A current sheet has K = 30 A m⁻¹. Find field magnitude on one side.

Hints

Use B ≈ μ₀nI for solenoid interior.
Toroid field scales as 1/r.
Use B = μ₀K/2 for sheet.

Answers

B = (4π × 10⁻⁷)(2000)(0.25) = 6.28 × 10⁻⁴ T.
It halves.
B = (4π × 10⁻⁷)(30)/2 = 1.88 × 10⁻⁵ T.

6) Summary + next steps

Ampere’s law gives fast, clean results when symmetry is strong.
Solenoids produce nearly uniform interior fields; toroids confine field around the core.
Always state the approximation limits (long, dense, or infinite geometry).

Next: Faraday’s Law Of Induction & Lenz’s Law Previous: Ampere’s Law Back To Electromagnetism