12.2 Magnetic Field Due to a Thin Straight Wire

Magnetic fields form around current-carrying wires, and their strength depends on current and distance from the wire. The Biot-Savart law gives a way to calculate these fields from small current elements.

The right-hand rule shows the direction of the magnetic field around a straight wire. The field forms concentric circles around the wire, with strength decreasing as you move farther away.

Magnetic Field of a Thin Straight Wire

Biot-Savart law for straight wires

• Calculates magnetic field $\vec{B}$ at a point in space due to current-carrying wire using $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$
  • $\mu_0$: permeability of free space ($4\pi \times 10^{-7} \text{ T} \cdot \text{m/A}$)
  • $I$: current flowing through wire (amperes)
  • $d\vec{l}$: infinitesimal length of wire (meters)
  • $\hat{r}$: unit vector pointing from wire segment to point in space
  • $r$: distance from wire segment to point in space (meters)
• Integrate Biot-Savart law over entire wire length for total magnetic field
  • Infinite straight wire: $B = \frac{\mu_0 I}{2\pi r}$
  • Finite straight wire of length $L$: $B = \frac{\mu_0 I}{4\pi r} (\sin \theta_2 - \sin \theta_1)$, $\theta_1$ and $\theta_2$ are angles between point and wire ends (radians)
• Ampère's law provides an alternative method for calculating magnetic fields around current-carrying wires

Current and distance in field strength

• Magnetic field strength directly proportional to current $I$ through wire
  • Doubling current doubles magnetic field strength
• Magnetic field strength inversely proportional to distance $r$ from wire
  • Doubling distance reduces magnetic field strength by factor of 2
  • Follows inverse square law for short wire segments, inverse linear relationship for infinite wires (falls off more slowly with distance)

Right-hand rule for wire fields

• Determines direction of magnetic field around current-carrying wire
  • Thumb points in current flow direction
  • Fingers curl around wire in magnetic field direction
• Magnetic field lines form concentric circles around wire
  • Circles lie in planes perpendicular to wire
  • Strongest field closest to wire, weakens with increasing distance
• Reversing current direction reverses magnetic field direction

Related Concepts

• Electromagnetic induction: process by which a changing magnetic field induces an electric current in a nearby conductor
• Magnetic flux: measure of the total magnetic field passing through a given area
• Solenoid: a coil of wire that produces a uniform magnetic field when current flows through it
• Magnetic dipole moment: a measure of the strength and orientation of a magnetic dipole (e.g., a current loop)
• Magnetic field lines: imaginary lines used to visualize the direction and strength of a magnetic field
• Magnetomotive force: the driving force for magnetic flux in a magnetic circuit