Magnetic Force Between Two Parallel Currents

B1

(1)

I1

(2)

r

(3)

I2

(4)

F2

(5)

Magnetic Force Between Current-Carrying Wires

Two current-carrying conductors exert magnetic forces on each other due to the magnetic fields produced by their currents.

B1 = ((μ0×I1)/(2πr))

The magnetic field produced by a long straight wire carrying current I1 at a distance r

F = I × l × B × sin(θ)

Mathematical representation of the force experienced by a wire of length l carrying a current of I in a magnetic field B.

sin(θ) = 1

For two parallel wires, the magnetic field is perpendicular to the current, therefore what’s the value of sin(θ)

F = I × l × B × 1

Magnetic Force on a Parallel Wire

F/l = ((μ0×I1×I2)/(2πr))

The magnetic force per unit length between two long, parallel currents I1 and I2 separated by distance r

Direction of Force Between Parallel Currents

Parallel currents in the same direction attract, while parallel currents in opposite directions repel.

Newton’s Third Law for Wire Forces

The magnetic forces that two current-carrying wires exert on each other are equal in magnitude and opposite in direction.

Right-Hand Rule for Wire–Wire Forces

The direction of the force on a wire is determined by applying the right-hand rule to the current direction and the magnetic field produced by the other wire.

Pinch Effect

In high-current arcs, magnetic attraction squeezes the current path, increasing temperature and potentially damaging equipment.

Pinch Effect in Electric Arcs

In high-current arcs, magnetic attraction squeezes the current path, increasing temperature and potentially damaging equipment.

Pinch Effect in Plasma Physics

Magnetic forces shape jets of ionized material, such as solar flares, by compressing plasma along current paths.

Role of Charge Neutrality in Magnetic Forces

Magnetic attraction dominates only when the net charge density is zero; otherwise, electric (Coulomb) repulsion overwhelms it.

Definition of the Ampere (Classical)

The ampere is defined based on the magnetic force between two long, parallel conductors carrying equal currents.

Standard Ampere Definition Condition

• Two parallel wires separated by 1 meter

• each carrying 1 ampere

• experience a force per unit length of:

• F/l = 2 × 10^-7 N/m

1T = 1 N/(A⋅m)

Relationship Between Tesla and Newton

Practical Measurement of Current

In practice, current is measured using a current balance that detects the magnetic force between nearby conductors

1C = 1 A⋅s

Definition of the Coulomb

Force Direction in Same-Direction Currents

When two wires carry current in the same direction, each wire is pulled toward the other.

Force Direction in Opposite-Direction Currents

When two wires carry current in opposite directions, the wires push away from each other.

Distance Dependence of Wire–Wire Force

The magnetic force per unit length decreases inversely with separation distance r.

Vector Nature of Magnetic Force Between Wires

The force has both magnitude and direction and must be expressed as a vector when wires are not collinear.

Geometry Dependence of Wire Forces

The magnitude and direction of the force depend on the relative positions of the wires, not just current magnitude.