12.3 Magnetic Force between Two Parallel Currents

Parallel current-carrying wires create magnetic fields that interact, causing attraction or repulsion. The force between them depends on current direction, magnitude, wire length, and separation. This interaction forms the basis for defining the ampere, the SI unit of electric current.

Understanding these forces is crucial for electrical engineering and physics applications. The relationship between current and magnetic fields showcases the fundamental connection between electricity and magnetism, a cornerstone of electromagnetic theory.

Magnetic Force between Parallel Currents

Magnetic interaction of parallel wires

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• Parallel current-carrying wires generate magnetic fields that interact with each other
  • Magnetic fields encircle the wires according to the right-hand rule (point thumb in current direction, fingers curl in field direction)
  • Field strength decreases with distance from the wire ($B \propto 1/r$)
• Interaction depends on relative current directions
  • Same direction currents attract due to opposite field directions between wires
  • Opposite direction currents repel due to same field directions between wires
• Force magnitude depends on current magnitudes, wire length, and separation distance
  • Proportional to product of current magnitudes ($F \propto I_1I_2$)
  • Proportional to wire length ($F \propto L$)
  • Inversely proportional to separation distance ($F \propto 1/r$)
• Force direction is always perpendicular to the plane containing the wires
  • Attraction pulls wires together perpendicular to their length
  • Repulsion pushes wires apart perpendicular to their length

Ampere definition using wire forces

• Ampere is the base SI unit for electric current
• Defined in terms of the force between two parallel wires
  • Consider two infinitely long, thin, parallel wires separated by 1 meter in vacuum
  • If equal currents in the wires produce an attractive force of $2 \times 10^{-7}$ N per meter of length, each current is defined as 1 ampere
• This definition connects the ampere to a measurable force between current-carrying wires
  • Provides a practical way to standardize the ampere based on a magnetic interaction
  • Highlights the relationship between electric current and magnetic fields

Force calculation for parallel wires

• The magnetic force per unit length between parallel current-carrying wires is given by:
  • $F/L = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{r}$
    • $\mu_0 = 4\pi \times 10^{-7}$ T⋅m/A is the permeability of free space
    • $I_1$ and $I_2$ are the currents in the two wires in amperes
    • $r$ is the separation distance between the wires in meters
• Multiply the force per unit length by the wire length to get the total force
  • $F_{total} = (F/L) \times L$
• Use the right-hand rule to determine the force direction
  • Point right thumb in the direction of the first current ($I_1$)
  • Curl fingers in the direction of the second current ($I_2$)
  • Extended index finger points in the direction of the force on the second wire
  • Reverse the roles of $I_1$ and $I_2$ to find the force direction on the first wire
• The force experienced by the wires is an example of the Lorentz force

Related Electromagnetic Concepts

• Magnetic dipole: The current loop in a wire creates a magnetic dipole, which interacts with external magnetic fields
• Magnetic flux: The amount of magnetic field passing through a given area, which can change due to the motion of parallel currents
• Electromagnetic induction: The process by which changing magnetic fields can induce currents in nearby conductors, including parallel wires
• Magnetic moment: A measure of the strength and orientation of a magnetic dipole, which affects the interaction between current-carrying wires