**Fedosin****'****s theorem**

**Fedosin's**** theorem **is a theorem on the magnetic field of rotating charged
bodies in classical electrodynamics. It was proven by Sergey Fedosin in 2021. ^{[1]}

The theorem
states that the magnetic field on the rotation axis of an axisymmetric charged
body or charge distribution has only one component directed along the rotation
axis, and the magnetic field is expressed through the surface integral, which
does not require integration over the azimuthal angle . In the general case, for arbitrary charge
distribution and for any location of the rotation axis, the magnetic field is
expressed through the volume integral, in which the integrand does not depend
on the angle .

**Contents**

·
1 Proof of theorem

·
2 Application

o
2.1 Cylinder

o
2.2 Ball

·
3 Significance of the
theorem

·
4 References

·
5 See also

·
6 External links

**Proof of theorem**

The starting point in the proof of the theorem is the electromagnetic Liénard–Wiechert potential. For the vector potential of a rotating charged
body, the following is obtained:

where is the
vacuum permeability; is the
invariant charge density of the matter; is
the Lorentz factor for the velocity of chaotic motion of matter charged
particles; is the
linear rotational velocity of an arbitrary point in the volume of the body at
an early moment of time ; is the
volume element of a non-rotating body in cylindrical coordinates; is the
distance taken at an early time from a point inside the body to a point with a
radius vector , where
the vector potential is sought; is the
constant angular velocity; is the
speed of light; is the
azimuth angle of an arbitrary point in the volume of the body, taken at the early
time.

The magnetic field is determined by the formula . When
calculating the rotor, it is necessary to find the partial derivatives , and from the
values , , and .

If the axis of rotation of the body is directed along the axis,
the magnetic field on this axis has only one component equal to

From this it can be seen that the integrand does not depend on the
azimuthal angle for any
position of the axis of rotation. For axisymmetric bodies, the formula for the
magnetic field on the axis of rotation is simplified and ceases to depend on
the angle :

**Application**

**Cylinder**

Formula (1) as applied to a uniformly charged rotating cylinder of
length and
radius , without taking into account the chaotic motion of charged
particles (), when the origin of the coordinate system is
at the center of the cylinder, gives the following for the field on the axis of
rotation outside
the cylinder:

At the end face of the cylinder at it turns
out

The magnetic field inside the cylinder on the
axis of rotation is:

At the center of the cylinder for the magnetic
field, we obtain

For a long cylinder with , the
magnetic field in the center is almost twice as large as that at the end of the
cylinder on the axis of rotation.

For
large , an approximate formula is valid:

**Ball**

Formula (1) for the magnetic field on the
rotation axis can be
written in spherical coordinates:

For a solid ball of radius , in
which the proper chaotic motion of charges is not taken into account, . If the
origin of the coordinate system is in the center of the ball, the formulas will
be valid for the external field on the axis of rotation and at the pole of the
ball:

When in the
center of the ball, the field is

**Significance
of the theorem**

The formulas found in the proof of the theorem
make it possible to determine the external magnetic field of rotating charged
bodies, as well as the field at their center. In addition, the results obtained
make it possible to calibrate the complete solutions for the magnetic field of
rotating bodies that satisfy the wave equations.

**References**

1.
Fedosin S.G.
The Theorem on the Magnetic Field of Rotating Charged Bodies. Progress In
Electromagnetics Research M, Vol. 103, pp. 115-127 (2021). __http://dx.doi.org/10.2528/PIERM21041203
.__ ArXiv __2107.07418__.
Bibcode __2021arXiv210707418F__.

**See**** also**

§
Electromagnetic field

§ Electromagnetic field of cylinder

**External**** links**

Source: http://sergf.ru/ften.htm