An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular component of a magnetic field is continuous across the interface of two magnetic mediums. In contrast, its parallel component, perpendicular to the current, is discontinuous by the amount equal to the product of the vacuum permeability and the surface current. Like the scalar potential in electrostatics, the vector potential is also continuous...

The magnetic field due to a volume current distribution given by the Biot–Savart Law can be expressed as follows:

The representation of magnetic fields by magnetic field lines is very useful in visualizing the strength and direction of the magnetic field. Each of the magnetic field lines forms a closed loop. The field lines emerge from the north pole (N), loop around to the south pole (S), and continue through the bar magnet back to the north pole.
Magnetic field lines follow several hard-and-fast rules:

Consider a circular loop with a radius a, that carries a current I. The magnetic field due to the current at an arbitrary point P along the axis of the loop can be calculated using the Biot-Savart law.

Two long, straight, and parallel current-carrying conductors exert a force of equal magnitude on one another. The direction of the force depends on the current direction in the conductors.
The force exerted by the magnetic field due to the first conductor over a finite length of the second conductor is given as the product of the current in the second conductor and  the vector product of the length vector along the current element and the field due to the first conductor. According to the...

The most common application of magnetic force on current-carrying wires is in electric motors. These consist of loops of wire, which are placed between the magnets with a magnetic field. When current flows through the loops, the magnetic field applies torque, which causes the shaft to rotate, thus converting electrical energy to mechanical energy.
Consider a rectangular current-carrying loop containing N turns of wire, placed in a uniform magnetic field. The net force on a current-carrying loop...