AcademySources of Magnetic Fields
Academy
Fields Around Straight Currents
Level 1 - Physics topic page in Sources of Magnetic Fields.
Principle
A long straight current produces circular magnetic field lines around the wire. The field strength decreases in inverse proportion to distance from the wire.
Notation
\(I\)
current in the straight wire
\(\mathrm{A}\)
\(r\)
perpendicular distance from the wire
\(\mathrm{m}\)
\(\vec B\)
magnetic field
\(\mathrm{T}\)
\(\mu_0\)
permeability of free space
\(\mathrm{N\,A^{-2}}\)
\(\hat\phi\)
azimuthal direction around the wire
Method
Derivation 1: Use circular symmetry
For a very long straight wire, every point the same distance from the wire has the same field magnitude. The direction is tangent to a circle centered on the wire.
Circular field direction
\[\vec B=B\hat\phi\]
Straight-wire magnitude
\[B=\frac{\mu_0 I}{2\pi r}\]
Derivation 2: Apply the right-hand grip rule
Point your right thumb along conventional current. Your fingers curl in the direction of the magnetic field around the wire.
Derivation 3: Add fields from several wires
Each wire produces its own circular field. At a chosen point, use the right-hand rule for each wire, then add the fields as vectors.
Rules
Long straight wire
\[B=\frac{\mu_0 I}{2\pi r}\]
Vector direction
\[\vec B=\frac{\mu_0 I}{2\pi r}\hat\phi\]
Superposition
\[\vec B_{\mathrm{net}}=\sum_i \vec B_i\]
Examples
Question
A vertical wire carries current upward. What is the magnetic field direction at a point to the right of the wire?
Answer
Point the right thumb upward. At the right side of the wire, the curled fingers point into the page.
Checks
- The formula assumes the wire is long compared with the distance from it.
- The magnetic field is tangent to circles centered on the wire.
- Doubling current doubles \(B\).
- Doubling distance halves \(B\).