Magnetic Force between Two Parallel Conductors (HL IB Physics)

• A current carrying conductor, such as a wire, produces a magnetic field around it
• The direction of the field depends on the direction of the current through the wire
  • This is determined by the right hand thumb rule

• Parallel current-carrying conductors will therefore either attract or repel each other
  • If the currents are in the same direction in both conductors, the magnetic field lines between the conductors cancel out – the conductors will attract each other
  • If the currents are in the opposite direction in both conductors, the magnetic field lines between the conductors push each other apart – the conductors will repel each other

Both wires will attract if their currents are in the same direction and repel if in opposite directions

• When the conductors attract, the direction of the magnetic forces will be towards each other
• When the conductors repel, the direction of the magnetic forces will be away from each other
• The magnitude of each force depends on the amount of current and the length of the wire

Force per Unit Length Between Two Parallel Conductors

• The ratio is the force per unit length between two parallel currents I₁ and I₂ separated by a distance r 
• The force is attractive if the currents are in the same direction and repulsive if they are in opposite directions
• It is calculated using the equation:

• Where:
• • F = the force applied between the two parallel wires (N)
  • L = the length of each parallel conductor (m)
  • μ₀ = the constant for the magnetic permeability of free space (4π × 10⁻⁷ N A⁻²)
  • I₁ = the current through the first conducting wire (A)
  • I₂ = the current through the second conducting wire (A)
  • r = the separation between the two conducting wires (m)

The forces on each of the current-carrying wires are equal and opposite in direction

Obtaining the Equation

• The force from wire 2 on wire 1, F₂ = B₂I₁Lsin(θ) 
• In this situation the magnetic field is perpendicular to the current in the wire, so sin(θ) = 1
• F₂ = −F₁ so the force between them is F
• The force on a unit length of the wires is then given by 
  • Hence, 
• The magnitude of the magnetic field at a radial distance, r away from the current conducting wire is:
  • In this case the magnetic field strength from B₂ at a distance r away from wire 2 is: 
• Substituting for B₂ into the force per unit length equation gives us: