Magnetic Force On A Current-Carrying Conductor

Moving charges experience a force in a magnetic field. Since the magnetic fields produced by moving charges are proportional to the current, a conductor carrying a current creates a magnetic field around it.

Consider a compass placed near a current-carrying wire. The wire experiences a force that aligns the needle of the compass tangentially around the wire. Thus, the current-carrying wire produces concentric circular loops of magnetic field. The magnetic field generated by a wire can be determined using the right-hand rule, which states that if the thumb points in the direction of the current, then the direction in which the fingers curl around the wire gives the direction of the magnetic field produced. Consider a rectangular plane in the XY direction. If the magnetic field moves out of the plane, those magnetic field lines are represented by a dot symbol. If the magnetic field moves into the plane, those magnetic field lines are represented by a cross symbol.

Consider a straight conducting wire with current flowing from the bottom to the top that has a length l and a cross-sectional area A. The conducting wire is placed in a uniform magnetic field that is perpendicular to the plane and directed into the plane. The drift velocity acts upward and is perpendicular to the magnetic field. The average force on each charge is directed to the left and is given as follows:

If there are n charges per unit volume, then the cross-sectional area of the wire multiplied by the number of charges per unit volume and the length gives the volume of the segmented wire. Thus, the total magnetic force on the segment has a magnitude as follows:

By recalling the drift velocity equation and substituting the terms, the total force on the segment can be calculated as follows: