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UY1: Ampere's Law

Apply Ampere's law with symmetry to derive magnetic fields and avoid common enclosed-current sign mistakes.

  • University Physics Year 1
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Learning objectives

  • Use vector and differential or integral calculus to express physical change and accumulation.
  • Use complex numbers, linear algebra, and differential equations to solve coupled physical models.
  • Choose an efficient mathematical method and validate the result physically.
Why this matters + quick links

This page gives the UY1 working model/result for 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.

1) At a glance

  • Magnetostatic (steady-current) Ampere’s law:
∮ vector B · d vector ℓ = μ₀ I_enc
  • Time-varying fix (Maxwell-Ampere): add displacement current when Φ_E changes (see Displacement Current).
  • Ampere’s law is most useful when symmetry makes vector B constant and parallel to d vector ℓ on your chosen loop (cylindrical, planar, toroidal symmetry).
  • Sign convention: loop direction ⇒ surface-normal by right-hand rule ⇒ I_enc sign.
  • Quick checks: B must be in tesla; for a long wire outside the conductor, B ∝ I and B ∝ 1/r.

Prerequisites: Magnetic Field Of A Current Element, Magnetic Field Of A Straight Current Carrying Conductor, Vector Calculus: Integral Theorems
Next uses: Applications Of Ampere’s Law, Displacement Current

2) Setup

Choose an Amperian loop that matches the field symmetry.

  • Direction of d vector ℓ sets positive normal direction.
  • I_enc is algebraic sum through the loop’s surface.
  • Magnetostatic model: currents are steady, fields do not change in time, and the displacement-current term is neglected.

For problems that explicitly involve time-varying electric fields (e.g. a charging capacitor), use Maxwell-Ampere law instead of the magnetostatic shortcut:

∮ vector B · d vector ℓ = μ₀(I_(C,enc) + ε₀dΦ_E/dt).
Right-hand rule + enclosed-current sign

Pick your orientation once and then stop changing it mid-solution.

  • Choose a direction for the loop integral ∮ d vector ℓ.
  • Curl your right-hand fingers with that direction; your thumb gives the positive surface normal n hat.
  • Currents piercing the surface in the n hat direction count positive; opposite direction counts negative.

3) Core derivation/explanation

General workflow:

  1. Use symmetry to infer direction of vector B.
  2. Pick loop where B is constant or zero on segments.
  3. Evaluate ∮ vector B · d vector ℓ.
  4. Set equal to μ₀I_enc.

Example result for long straight wire:

  • Circular loop radius r gives vector B∥ d vector ℓ and constant B.
∮ vector B · d vector ℓ = B(2π r) = μ₀I ⇒ B = μ₀I/(2π r).

For a solid wire radius R with uniform current density:

  • Inside (r < R): I_enc = I(r²/R²)
B(2π r) = μ₀Ir²/R² ⇒ B = μ₀Ir/(2π R²).
  • Outside (r ≥ R): recover B = μ₀I/(2π r).

Checks (sanity + sign)

  • Continuity at the surface: for uniform current density, the inside and outside formulas agree at r = R.
  • Limiting cases: as r → 0, the uniform-wire inside result gives B → 0; as r increases outside, B decays like 1/r.
  • Direction: around a straight wire, vector B must be tangent to circles centered on the wire axis (right-hand grip rule with current).

4) Worked example(s)

A long solid conductor has radius R = 1.0 cm and total current I = 12 A uniformly distributed.

Find B at r = 5.0 mm (inside):

B = μ₀Ir/(2π R²) = ((4π × 10⁻⁷)(12)(5.0 × 10⁻³))/(2π(1.0 × 10⁻²)²) = 1.2 × 10⁻⁴ T.

5) Practice set (with hints + answers)

  1. Why is Ampere’s law most useful only in high-symmetry cases?
  2. A long wire carries I = 6.0 A. Find B at r = 3.0 cm.
  3. In a chosen loop orientation, one current passes out of surface and one equal current passes into surface. What is I_enc?

Hints

  • The law is always true, but symmetry makes the integral solvable.
  • Use B = μ₀I/(2π r) for long wire.
  • Enclosed current is algebraic sum with signs.

Answers

  1. Without symmetry, B is not constant along the loop and integral is hard to evaluate directly.
  2. B = 4.0 × 10⁻⁵ T.
  3. I_enc = 0.

6) Summary + next steps

  • Ampere’s law is a circulation statement for magnetic fields.
  • Success depends on good loop choice and sign discipline for enclosed current.
  • It quickly reproduces classic fields for wires and current distributions.

Next: Applications Of Ampere’s Law Previous: Magnetic Field Of A Circular Current Loop Back To Electromagnetism