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NEOCLASSICAL THEORY OF INTERACTION 49

come to the conclusion that the current loop will spin clockwise around its axis the same way
as revolving a door in Figure 2.1.1b. Subsequently, defining the torque as the cross product of
twisting force and the distance from the point of the force application we obtain for the magnetic

torque volume density applied to each side of the loop or half of total torque

⁄
= x 2 = x x /2 (2.1)

Assuming the vector of current density and magnetic field

B do not vary along the whole loop we have after
transformations
= 2 ∫ = x [W⋅s] (2.2)

Here = , is the wire cross sectional area and dl is
the infinitesimal wire length, = is the area of the loop
rectangle, and is the unit vector normal to this rectangle.
Then the vector called the magnetic moment of small

loop with steady current is shown in Figure 2.1.3 and
= (2.3)

If so, the torque is

= x (2.4) Figure 2.1.3 Torque a)

maximum, b) zero
This equity includes three vital components (external
magnetic field B, electric current , and any loop characterized by its area A) of electrical

motors, a moving force of our civilization. According to some estimations, electric motors of
all types consume from 25% to 50% electricity used by human beings. The history of an electric
motor, probably, started in 1822 when British physicist and mathematician Peter Barlow built
the first rotating device (Barlow's Wheel) driven by electromagnetic forces. The first patent for
DC electric motor (run from any kind of battery or DC power supply) was granted in 1837 to
American inventor Thomas Davenport. The first real AC motors were invented by Nikola Tesla
in the 1880s. Modern electric motors are extremely efficient converting more than 90% of
electrical energy to kinetic energy and their progress based on experimental, and numerical
approaches are the fascinated and well-established chapter of electromagnetic theory. We are
not going to pursue this subject further sending the reader for details to the specialized literature
[11].

From above discussion and equity (2.4) follows that the twisting force of torque reaches the
maximum when ⊥ and completely demises when ∥ , as shown in Figure 1.2.3a
and b. Therefore, the rotation stops when the loop magnetic moment fully aligns with the
external magnetic field vector or the plane of loop is perpendicular to vector B. Consequently,
the total magnetic field strength as a sum of external and current loop field increases. As soon
as the magnetic moment lines up with the magnetic field, the potential energy is at its lowest,
i.e. loop orientation is at its most stable position. If so, some additional energy should be apply
to re-align the loop located in an external magnetic field.
We will demonstrate later how similar effect explains the phenomena of magnetic polarization
in materials.