Magnetic fields around wires are a key concept in electromagnetism. The Biot-Savart law helps us calculate these fields, showing how they depend on current and distance. Understanding this relationship is crucial for many electrical applications.

The right-hand rule is a handy tool for visualizing magnetic fields around wires. It shows how the field forms concentric circles around the wire, with strength decreasing as you move farther away. This knowledge is essential for working with electromagnets and motors.

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