Force Between Two Parallel Current-Carrying Conductors : Definition of Ampere

When two long, straight conductors carrying electric current are placed parallel to each other, each conductor produces a magnetic field that exerts a force on the other. This mutual magnetic interaction is the physical basis for the official SI definition of the ampere.

The magnetic field at a perpendicular distance r from a long straight conductor carrying current I₁​ is given by:

\[B_1=\frac{\mu_0I_1}{2\pi r}\]

A constant representing the ability of vacuum to support magnetic field lines; its value is 4π × 10⁻⁷N A⁻².

One ampere is the value of steady current which, when maintained in each of two infinitely long, straight, parallel conductors of negligible cross-section, placed 1 metre apart in vacuum, produces a force of 2 × 10⁻⁷ newton per metre of length between them.

Step 1: Consider two long, straight, parallel conductors, separated by a distance r, carrying currents I₁ and I₂, respectively.

Step 2: The magnetic field produced by conductor 1 at the location of conductor 2 is: \[B_1=\frac{\mu_0I_1}{2\pi r}\]

Step 3: This field exerts a force on a length L of conductor 2 carrying current I₂​, using F = B₁I₂L: F = ​\[\frac{\mu_0I_1I_2L}{2\pi r}\]

Step 4: Dividing both sides by L, the force per unit length is: \[\frac{F}{L}=\frac{\mu_0I_1I_2}{2\pi r}\]

This definition directly follows from the force formula by substituting I₁ = I₂ = 1 A, r = 1 m, and μ₀ = 4π × 10⁻⁷N A⁻²:

Think of each current-carrying wire as a tiny bar magnet. Two bar magnets with like poles facing each other repel, while unlike poles attract. Similarly, wires carrying current in the same direction behave like attracting magnets, while opposite-direction currents behave like repelling ones.