Magnetic force on a conductive wire

Whenever a charge is put under the influence of a magnetic field, it undergoes an interaction that can alter its motion. If the magnetic field in question is uniform, we have seen that there will be a force acting on the charge with intensity , Where is the angle formed in the plane between the velocity vectors and the magnetic field. The direction and direction of the vector will be given by the flat right hand rule.

If we imagine a current-carrying conductor wire, there will be free electrons moving through its cross section at a speed . However, the direction adopted for the velocity vector, in this case, is the actual direction of the current ( has the same sense of current). For ease of understanding one can imagine that free electrons are positive charges.

As all free electrons have charge (which by the assumption adopted behaves as if it were positive), when the conductor wire is exposed to a uniform magnetic field, each electron will be acted upon by a magnetic force.

But if we consider a small piece of wire rather than just an electron, we can say that the interaction will continue to be governed by where Q is the total charge in the wire segment, but since we have a length traveled by each electron over a given time interval, then we can write the velocity as:

By replacing this value in we will have the magnetic force in the segment, expressed by the notation :

But we know that indicates the current intensity in the wire, then:

Being this expression called the Laplace Elemental Law.

The direction and direction of the vector are perpendicular to the plane determined by the vectors and , and can be determined by the rule of the flat right hand, pointing the thumb in the direction of the current and the other fingers in the direction of the vector. .

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If we want to determine the magnetic force acting on long wire (with non-negligible dimensions) we must make the lengths smaller and smaller and sum the vectors in each , so that the whole wire is described, an advanced way to perform this calculation is to use line integral.

For the particular case where the conductor is rectilinear, all vectors will be equal, so we can rewrite the elementary law of Laplace as .