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UY1: Magnetic Field Of A Straight Current Carrying Conductor

Find magnetic field around a long straight conductor and apply right-hand-rule direction and inverse-distance scaling.

  • University Physics Year 1
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Learning objectives

  • Analyse magnetic forces, induction, inductance, and alternating-current systems with consistent signs.
Why this matters + quick links

This page gives the UY1 working model/result for Magnetic Field Of A Straight Current Carrying Conductor. 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

  • For a long straight wire:
B(r) = (μ₀ I)/(2π r)
  • Field lines are concentric circles around the wire.
  • Direction comes from right-hand grip rule (thumb along current).
  • Modelling context: “long” means your observation distance r is small compared with the wire length, so end effects are negligible.

Prerequisites: Magnetic Field Of A Current Element, Ampere’s Law
Next uses: Magnetic Field & Force Between Parallel Conductors

2) Setup

Assume a very long straight conductor carrying steady current I.

  • Observation point is distance r from wire axis.
  • Symmetry implies same B magnitude on a circle of radius r.
  • SI units: B in tesla, r in metres.
Common traps (right-hand rule + what r means)
  • Use conventional current direction for the right-hand grip rule. Electron drift is opposite to conventional current.
  • The distance r is measured from the wire axis, not from the wire surface (unless you are explicitly given a wire radius and asked for inside/outside behavior).
  • If current reverses, the field direction reverses but the magnitude formula stays the same.

3) Core derivation/explanation

From symmetry and Ampere’s law (or Biot-Savart integration), the magnetic field magnitude at radius r is:

B = (μ₀ I)/(2π r).

Interpretation:

  • B ∝ I: stronger current gives stronger field.
  • B ∝ 1/r: farther points have weaker field.

Direction check:

  • Current upward: field circles anticlockwise when viewed from above.
  • Current downward: direction reverses.

Checks (sanity)

  • B decreases with increasing r (inverse-distance law).
  • If I → 0, then B → 0.

4) Worked example(s)

A long wire carries I = 8.0 A. Find B at r = 4.0 cm.

B = (μ₀ I)/(2π r) = ((4π × 10⁻⁷)(8.0))/(2π(4.0 × 10⁻²)) = 4.0 × 10⁻⁵ T.

So the field is 40 μ T, about Earth’s-field scale.

5) Practice set (with hints + answers)

  1. If r is tripled with same I, what is new B/B₀?
  2. A wire carries 20 A; find B at r = 0.10 m.
  3. If current reverses, what happens to B magnitude and direction?

Hints

  • Use inverse proportionality in r.
  • Substitute directly into B = μ₀ I/(2π r).
  • Direction from right-hand rule.

Answers

  1. B/B₀ = 1/3.
  2. B = 4.0 × 10⁻⁵ T.
  3. Magnitude unchanged; direction reverses.

6) Summary + next steps

  • Long straight-wire fields are circular and scale as I/r.
  • Direction logic is essential for superposition problems.
  • This result is the building block for forces between conductors.

Next: Magnetic Field & Force Between Parallel Conductors Previous: Magnetic Field Of A Moving Charge Back To Electromagnetism