**Electromagnetic field of
cylinder**

In classical electromagnetism, **electromagnetic field of cylinder** is
considered as the field of one of the simplest geometric bodies. The solutions
for the components of the electromagnetic field for the case of a stationary
and rotating cylinder are not much more complicated than the corresponding
solutions for a ball.

When a uniformly charged cylinder rotates with a
constant angular velocity, the field is stationary and does not depend on time.
In this case, for electric scalar potential
and for magnetic vector
potential from Maxwell's
equations follow equations:

where
is the Laplace operator;
is the Lorentz factor;
is the invariant charge density of the cylinder matter;
is the electric constant;
is the speed of light;
is the linear speed of rotation of an arbitrary point taken in the volume of
the cylinder.

**Содержание**

·
1Long
stationary cylinder

·
2Long rotating
cylinder

o
2.1Vector
potential and magnetic field

·
3References

·
4See also

·
5External links

**Long stationary cylinder**

In a fixed cylinder, the Lorentz factor of charged
particles of matter is
, if the proper chaotic motion of these particles is not taken into account. It
is convenient to express the Laplacian in (1) in cylindrical coordinates
. In a sufficiently long cylinder, one
can neglect the edge effects that are significant near the ends of the
cylinder, and assume that the field mainly depends only on the coordinate
. In this approximation, the electric potential and the electric field strength
inside the cylinder are equal: ^{[1]}

where
is the length of the cylinder,
is the radius of the cylinder, is the unit vector along
the cylindrical coordinate .

As it can be seen, the potential inside the cylinder
depends on its length
logarithmically, due to the presence of the inverse hyperbolic sine
. The internal electric field
far from the ends of the cylinder at
is directed perpendicular to the axis of rotation and is equal to zero on the
axis of rotation, where .

The corresponding external electric potential and
electric field strength outside the long cylinder are as follows:

The above formulas require correction near the ends of
the cylinder, since here the electric potential and field strength become
functions not only of
, but also of .

**Long rotating cylinder**

When the cylinder rotates with a constant angular
velocity
, the Lorentz factor of charged particles of matter becomes the function of :

Taking this into account, the solution to Eq. (1) for the
scalar potential, as well as for the field strength inside a rotating uniformly
charged cylinder far from the ends of the cylinder will be as follows: ^{[1]}

Outside a long rotating cylinder, the scalar potential
and electric field strength are expressed by the formulas:

**Vector potential
and magnetic field**]

The rotation of the charged matter of the cylinder
leads to the appearance of the vector potential
and the induction of the magnetic field
. For these values inside the cylinder, far from the ends of the cylinder, as a
result of (2), the following is obtained:

where is a unit vector directed
along the cylindrical coordinate
,
is a unit vector directed along the cylindrical coordinate .
As it can be seen, the internal vector potential rotates around the axis of
rotation of the cylinder. As for the magnetic field, it is directed along the
rotation axis along which the
coordinate is measured. In this case, the magnetic field is maximum on
the axis itself and tends to zero near the surface of the cylinder.

The external vector potential and the magnetic field
of a long cylinder are determined by the formulas:

These formulas are reasonably accurate near the center
of a long cylinder. However, as one approaches the ends of the cylinder, one
should take into account the fact that significant additions appear in the
formulas for the vector potential and magnetic field due to the dependence on
the
coordinate. For an infinitely long cylinder, the above formulas can be used
without restrictions.

Fedosin's theorem makes it
possible to accurately calculate the magnetic field on the axis of rotation of
charged rotating bodies. In particular, the magnetic field inside the cylinder
depends on
: ^{[2]}

At the center of the cylinder at
, the magnetic field is:

If we take points on the axis of rotation outside the
cylinder, then the magnetic field there looks like:

At the end of the cylinder at
, we get

As a result, the magnetic field at the center is
almost twice as large as at the end of the cylinder on the axis of rotation.
This difference shows the degree of influence of edge effects and the need to
take into account in (2) the dependence of the vector potential on the
coordinate
near the ends of the cylinder.

**References**

1.
^{1.0} ^{1.1} Sergey
G. Fedosin. The Electromagnetic
Field of a Rotating Relativistic Uniform System. Chapter 2 in the book:
Horizons in World Physics. Volume 306. Edited by Albert Reimer, New York, Nova
Science Publishers Inc, pp. 53-128 (2021), ISBN: 978-1-68507-077-9,
978-1-68507-088-5 (e-book). https://doi.org/10.52305/RSRF2992. // Электромагнитное поле
вращающейся релятивистской однородной системы.

2.
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. // Теорема о магнитном
поле вращающихся заряженных тел.

**See**** ****also**

·
Electromagnetic field

**External**** ****links**

·
Electromagnetic
field of cylinder in Russian

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