No, it was found by expirement.

Depends on whether a path integral or a normal integral is easier to determine.

According to Coulomb's law, the electrostatic force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of distance between their centers.

And Biot Savart's law gives the magnetic field at point due to a small current element. And, according to Biot Savart's law ,the magnitude of magnetic field is directly proportional to the product of current element Idl and Sin theta and inversely proportional to the square of distance between two point charges.

A classic and ancient experiment is to get a compass, a battery (9volt is fine), and some wire. Notice that the compass changes when the circuit is closed (wire connected to the two terminals). You can also see that the compass gets affected less when it is far from the wire versus next to it.

If your interested in the math behind the experiment, you may want to do some research on the Biot-Savart Law. The Biot-Savart law describe the magnetis field when the current is constant, or not time-varying. A more general form of the equation is called Ampere's Law, and the more general case of that is Maxwell's equations.