Two Paradoxes of the Existence of magnetic Fields
Published in physic.philica.com
The paradoxes of magnetic fields presented here, show some similarity with two paradoxes of electrical charges and fields, which the author presented in .
Outline of the Article
The first paradoxon of the magnetic field is explained in the first three sections:
(1.) Calculation of the field strength of the magnetic field of a moving electrical charge
Let us perform our calculations with the following example: The moving electrical charge which produces the magnetic field, shall be geometrically arranged homogeneously in a line with infinite length, orientated along of the z-axis, and the whole line is moving continuously in z-direction with constant speed, as illustrated in fig.1. The absolute value of the magnetic field strength can than be found in a usual standard textbook for students, for instance as  or . It is
with I = electrical current and r = (x2+y2)1/2.
Its energy density u can be found in the same textbooks. It is
Fig. 1: Illustration of the configuration of the moving electrical charge of our example, which produces the same magnetic field as a straight conductor with infinite length, With orientation of the conductor along the z-axis, the absolute value of the magnetic field strength can easily be given in cylinder coordinates according to equation (1).
(2.) Propagation of the magnetic field respectively of its energy through the space
Now we want to find out, how much magnetic field energy is flowing into the space within a time interval . Therefore we adjust the time-scale as following: The electrical current (i.e. the movement of the electrical charge) is switched on in the moment . The time (with ) shall be defined as the moment, at which the magnetic field reaches the radius in consideration of its finite speed of propagation. Again a bit later, namely at the moment (with , the field will reach a cylinder with a radius . Consequently the magnetic energy, which has been emitted by the moving charge within the time interval , has to be the same energy, which fills the cylindrical shell from the inner radius up to the outer radius . We calculate this amount of energy by integration of the energy density inside the cylindrical shell, following equation (2) in which we introduce the magnetic field according to (1):
The time interval , within which this amount of energy has been emitted, can be determined from the speed of propagation of the magnetic field, which is the speed of light, as we know from the mechanism of the Hertz’ian dipole emitter . It must be the same speed, with which also the electrical field propagates. Thus the time interval can be calculated as following:
Now we can also write and as a function of time:
(3.) Calculation of the emitted power
In order to find out, whether the power P is constant in time (as it should be expected, because the charge keeps constant speed), we have to express the radii and as a function of time. For this purpose we can use the equation (5) and (6) and (7):
Obviously this expression is not constant in time. If it would be constant in time, it could not depend on the time , because it should be always the same for a given , independent on the moment of observation . This explains the first paradoxon of the magnetic field as mentioned above: The moving charge produces a magnetic field and with it, it emanates power, but the charge itself does not alter its own energy at all, because it keeps constant speed. Paradox is the fact, that we can not see the origin of the energy emitted by the moving charge. And the first part of the paradoxon of the magnetic field has a further aspect: The emitted power is not constant in time, but we do not have any indication for an alteration of the emitted field strength.
Let us now come to the second paradoxon:
? At the moment our cylinder had had the inner radius and the outer radius , this means that we look to the same cylindrical shell as in the sections 1…3.
? At the moment this cylindrical shell from to has propagated radially into the space until it reaches the inner radius of and the outer radius of .
? At it contains the energy
? At it contains the energy
This means, that we found the energy to be more than .
So, our volume element had lost energy just by propagating into the space. This unexplained loss of energy can be understood as a second paradoxon of the magnetic field. We remember that our magnetic field is static (in the given frame of reference), this means that at every position which the field already reached, the field strength will remain constant during time. And the speed with which the field producing charges moves is also constant during time, beginning with the moment , at which the electric current had been switched on.
The open question, which comes from both of these magnetic paradoxes is the following: From where does the energy of the magnetic field come, which is emanated by the moving charge, and where does the field energy go, when the field emanates into the space ?
1. Turtur, Claus W. (2007). Two Paradoxes of the Existence of electric Charge. arXiv:physics/0710.3253v1
2. Jackson, John David (1981). Klassische Elektrodynamik. Walter de Gruyter Verlag, Berlin, Germany, ISBN 3-11-007415-X
3. Giancoli, Douglas C. (2006). Physik. Pearson Studium, ISBN-13: 978-3-8273-7157-7
4. Gobrecht, Heinrich, et.al. (1971) Bergmann Schaefer - Lehrbuch der Experimentalphysik, vol.2, Walter de Gruyter Verlag, Berlin, Germany, ISBN 3-11-002090-0
Adress of the Author:
Prof. Dr. Claus W. Turtur
University of Applied Sciences Braunschweig-Wolfenbuettel
Salzdahlumer Strasse 46 / 48
Germany - 38302 Wolfenbuettel
Information about this Article
Published on Wednesday 19th December, 2007 at 09:31:51.
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