11.4 Magnetic Force on a Current-Carrying Conductor

Current-carrying wires create magnetic fields, forming concentric circles around the wire. The field strength depends on current magnitude and distance. Ampère's law relates the magnetic field around a closed loop to the electric current passing through it.

Magnetic forces on current-carrying wires in uniform fields are calculated using F = ILB sin θ. The right-hand rule determines force direction. Magnetic flux, permeability, and dipole moments are key concepts in understanding magnetic properties and interactions.

Magnetic Fields and Forces

Magnetic fields from current-carrying wires

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• Current flowing through a wire creates a magnetic field surrounding the wire
  • Magnetic field lines form concentric circles around the current-carrying wire (solenoid, electromagnet)
  • Right-hand rule determines the direction of the magnetic field
    • Point thumb in the current direction and fingers will curl in the magnetic field direction (battery, generator)
• Magnetic field strength is affected by the current magnitude and distance from the wire
  • Directly proportional to the current: doubling the current doubles the magnetic field strength
  • Inversely proportional to the distance: magnetic field weakens as distance from the wire increases (power lines, transformers)
• Ampère's law relates the magnetic field around a closed loop to the electric current passing through the loop

Force calculation for wires in magnetic fields

• Magnetic force on a current-carrying wire in a uniform magnetic field is calculated using $F = ILB\sin\theta$
  • $F$: magnetic force measured in newtons (N)
  • $I$: current flowing through the wire in amperes (A) (circuit, appliance)
  • $L$: length of the wire segment in meters (m)
  • $B$: magnetic field strength in teslas (T) (MRI machine, particle accelerator)
  • $\theta$: angle between the current direction and the magnetic field
• Maximum force occurs when the current is perpendicular to the magnetic field ($\theta = 90^\circ$)
  • $\sin 90^\circ = 1$, simplifying the equation to $F = ILB$
• No force is exerted when the current is parallel to the magnetic field ($\theta = 0^\circ$ or $180^\circ$)
  • $\sin 0^\circ = \sin 180^\circ = 0$, resulting in $F = 0$ (compass needle, magnetic levitation)
• The Lorentz force describes the force experienced by a charged particle moving in a magnetic field

Right-hand rule for magnetic force direction

• Right-hand rule determines the direction of the magnetic force on a current-carrying wire
  1. Point fingers in the current direction
  2. Orient palm to face the magnetic field direction
  3. Thumb will point in the magnetic force direction (electric motor, loudspeaker)
• Magnetic force is always perpendicular to both the current and the magnetic field
  • Force is a cross product of the current and magnetic field vectors (Hall effect sensor, velocity selector)
• Reversing either the current or magnetic field direction will reverse the force direction
  • Allows for precise control of magnetic forces in applications (electromagnetic crane, particle steering)

Magnetic Properties and Interactions

• Magnetic flux represents the amount of magnetic field passing through a given area
• Permeability is a measure of a material's ability to support the formation of a magnetic field within itself
• The magnetic dipole moment characterizes the torque experienced by a magnet in an external magnetic field