Suppose two charged conducting wires each have current I, and they're placed parallel to each other. Fins the line charge density that needs to be placed on each wire so that the electric repulsion between them balances the magnetostatic force so that they are in equilibrium.

To find the line charge density on each wire, we need to balance the electric repulsion and the magnetostatic force between them. Let's break it down step by step.

Step 1: Electric Repulsion
The electric repulsion between the two wires is given by Coulomb's Law, which states that the force between two charged objects is proportional to the product of their charges and inversely proportional to the square of the distance between them.

In this case, since we have infinitely long wires parallel to each other, we can treat them as line charges. The force between these line charges depends on the line charge density (λ) and the length of the wires (L). The electric repulsion force (Fe) between them is given by:

Fe = k * λ^2 / (2πd)

where k is the Coulomb's constant and d is the distance between the wires.

Step 2: Magnetostatic Force
The magnetostatic force between the wires is caused by the interaction of their currents. When two current-carrying wires are placed parallel to each other, a magnetic field is generated around each wire. These magnetic fields interact and create a force between the wires.

The magnetostatic force (Fm) between two parallel wires is given by Ampere's Law:

Fm = μ0 * I^2 * L / (2πd)

where μ0 is the permeability of free space, I is the current in each wire, and L is the length of the wires.

Step 3: Equilibrium Condition
To achieve equilibrium, the electric repulsion force must balance the magnetostatic force. Therefore, we equate Fe and Fm:

k * λ^2 / (2πd) = μ0 * I^2 * L / (2πd)

Simplifying the equation:

λ^2 = μ0 * I^2 * L / k

Taking the square root of both sides:

λ = √(μ0 * I^2 * L / k)

Finally, we have the expression for the line charge density (λ) required on each wire to achieve equilibrium.

Note that the sign of the line charge density will depend on the direction of current flow and determine whether the wires attract or repel each other.

It's important to understand that this is a derived equation based on certain assumptions and conditions, and it may not apply to all scenarios.