Encyclopedia of Electrical Engineering

We know that a current-carrying conductor creates a circulatory magnetic field around it. Whenever this current-carrying conductor of some length is placed perpendicular to a magnetic field, force is exerted on the conductor to move it in a direction perpendicular to the flow of current and that of the magnetic field.
Magnetic force on the current-carrying conductor and magnetic field around a conductor has been discussed. Here we directing our focus on formulating force between two parallels current carrying conductors. Because we already know, force on a single current carrying conductor and magnetic field around it. If we combine the equation for the strength of the magnetic field surrounding a current carrying conductor with the equation for the force on a current carrying conductor due to a magnetic field we can derive a formula for the force between two current carrying conductors, conductor 1 and conductor 2, with currents $I_1$ and $I_2$ (A) respectively, with length $l$ (m) a distance $d$ (m) apart.
Substituting into our equation for the force on a conductor due to a magnetic field for B we can derive an equation for the force on conductor 2, carrying current $I_2$, due to the magnetic field of conductor 1 carrying a current of $I_1$:
$$F=I_2.l.k {I_1 \over d} \sin\phi$$
as the two conductors are parallel $\sin\phi = 1$.

Therefore $$F=I_2.l.k {I_1 \over d}$$ Rearranging; $$\bbox[5px,border:1px solid red] {\color{blue}{{F \over l} =k{I_2 I_1 \over d}}} \tag{3}$$#### Direction of the force between two parallel current-carrying conductors

The magnetic fields of two straight parallel wires will interact to produce an attractive force if the currents in the wires are in the same direction, and a repulsive force if the currents in the wires are in opposite directions.
The direction of the force between two parallel conductors is the result of the direction of the magnetic field generated by each conductor, which is clockwise around the wire when looking along the wire in the direction of current flow. In areas where the magnetic fields are oriented in the same direction the overall magnetic field strength is reinforced, and in regions where the magnetic fields are in opposite directions the overall magnetic field strength is reduced.
The magnetic fields of the conductors interact to produce a force, oriented in such a way as to either push the wires apart, or pull them together, as required to create a uniform magnetic field around each conductor.

Therefore $$F=I_2.l.k {I_1 \over d}$$ Rearranging; $$\bbox[5px,border:1px solid red] {\color{blue}{{F \over l} =k{I_2 I_1 \over d}}} \tag{3}$$