Fields Around Parallel Conductors

Fields Around Parallel Wires

Fields Around Parallel Wires

• 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

Repulsion & Attraction of Current-Carrying Wires

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

• $\frac{F}{L}$ 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:

$\frac{\mathbf{F}}{\mathbf{L}}\mathbf{ }\mathbf{=}\mathbf{ }{\mathit{\mu }}_{\mathbf{0}}\mathbf{ }\frac{{\mathbf{I}}_{\mathbf{1}}{\mathbf{I}}_{\mathbf{2}}}{\mathbf{2}\mathbf{\pi }\mathbf{r}}$

• Where:

  • F is the force applied between the two parallel wires (N)

  • L is the length of each parallel conductor (m)

  • μ₀ is the constant for the magnetic permeability of free space = 4π × 10⁻⁷ N A⁻²

  • I₁ is the current through the first conducting wire (A)

  • I₂ is the current through the second conducting wire (A)

  • r is the separation between the two conducting wires (m)

Forces on Current-Carrying Wires

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 $\frac{F}{L} = \frac{B_2{I}_{1}L}{L}$

  • Hence, $\frac{F}{L} = B_2{I}_1$

• The magnitude of the magnetic field at a radial distance, r away from the current conducting wire is: $B =\frac{ {\mu }_{0}I}{2\pi r}$

  • In this case the magnetic field strength from B₂ at a distance r away from wire 2 is: $B_2 = \frac{{\mu }_0{I}_2}{2\pi r}$

• Substituting for B₂ into the force per unit length equation gives us: $\frac{F}{L} = \left(\frac{{\mu }_0{I}_2}{2\pi r}\right)I_1$