To find the electric force of interaction between two infinitely long straight wires with linear charge densities λ1 and λ2, we can use Coulomb's Law.


Coulomb's Law states that the electric force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically, it can be expressed as:

F = k * (q1 * q2) / r^2

Where:
F is the electric force between the two charges,
k is the electrostatic constant (k = 8.99 x 10^9 Nm^2/C^2),
q1 and q2 are the charges of the two objects, and
r is the distance between the two charges.

In the case of two infinitely long straight wires, we need to consider the interaction of each wire with the other wire.


To find the electric force on wire 1 due to wire 2, we need to consider a small segment of wire 1 and calculate the force between that segment and wire 2. We can then integrate this force over the entire length of wire 1 to find the total force.

- Consider a small segment of wire 1 with length dl and charge dq = λ1 * dl.
- The distance between the segment of wire 1 and wire 2 is d.
- Using Coulomb's Law, the electric force between the segment of wire 1 and wire 2 is dF = k * (dq * λ2) / d^2.
- The total force on wire 1 due to wire 2 can be obtained by integrating this force over the entire length of wire 1.


Similarly, we can calculate the electric force on wire 2 due to wire 1 by considering a small segment of wire 2 and integrating the force over the entire length of wire 2.


The net electric force between the two wires is the vector sum of the forces on each wire due to the other wire. Since the wires are perpendicular to each other, the forces will also be perpendicular. Therefore, the net force can be calculated using the Pythagorean theorem.

- The net electric force between the two wires is given by F_net = sqrt(F1^2 + F2^2), where F1 is the force on wire 1 due to wire 2 and F2 is the force on wire 2 due to wire 1.

In conclusion, to find the electric force of interaction between two infinitely long straight wires with linear charge densities λ1 and λ2, we need to calculate the forces on each wire due to the other wire and then find the vector sum of these forces.