Abstract
The Maxwell equations accounted for all the electricity and magnetism phenomena and predicted the electromagnetic waves but were in contrast with the Galileian relativity.
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- 1.
For two inertial frames \(S^{\prime }\) and S related by a Galileian transformation:
$$ x^{\prime }=x-vt \qquad y^{\prime }=y \qquad z^{\prime }=z \qquad t^{\prime }=t $$$$ x=x^{\prime }+vt^{\prime } \qquad y=y^{\prime } \qquad z=z^{\prime } \qquad t=t^{\prime } $$$$ {{\partial ~}\over {\partial x^{\prime }}}={{\partial ~}\over {\partial x}} \qquad {{\partial ~}\over {\partial t^{\prime }}}={{\partial ~}\over {\partial t}}+v{{\partial ~}\over {\partial x}} $$$$ {{\partial ^2~}\over {\partial {x^{\prime }}^2}}-{1 \over c^2}{{\partial ^2~}\over {\partial {t^{\prime }}^2}}=0 \quad \rightarrow \quad \left( 1-{v^2 \over c^2}\right) {{\partial ^2~}\over {\partial x^2}}-{1 \over c^2}{{\partial ^2~}\over {\partial t^2}}-2v{{\partial ~}\over {\partial x}}{{\partial ~}\over {\partial t}}=0\,. $$ - 2.
For more details see for instance J.D. Jackson: Classical Electrodynamics, cited, Chap. 11.
- 3.
In the frame of the magnet the electromotive force can be explained with the Lorentz force acting on the charges inside the loop and moving with the loop in the field of the magnet. In the frame of the loop the Lorentz force cannot act because the loop is at rest and the electromotive force is associated to the rate of change of the magnetic flux enclosed by the circuit. In relativity this incoherence is removed because an electric field arises from the transformation of the magnetic field of the magnet to the loop reference frame.
- 4.
A. Einstein: On the Electrodynamics of Moving Bodies.
- 5.
This subject can be found in many Electrodynamics textbooks as for instance L.D. Landau-E.M. Lifšits, The classical theory of fields, Chapters III and IV, or J.D. Jackson, Classical Electrodynamics, cited, Chap. 11. In these two books with an introduction to the theory of relativity, the Kinematics and the Dynamics are also considered.
- 6.
In this simple case the Lorentz transformations are:
$$ ct^{\prime }=(ct-\beta x)\gamma \qquad x^{\prime }=(x-vt)\gamma \qquad y^{\prime }=y \qquad z^{\prime }=z \qquad \beta ={v \over c} \qquad \gamma = {1 \over {\sqrt{1-\beta ^2}}}.$$ - 7.
The vector equation exists by itself in the space and of course it is independent from the frame. When considering the equation in a frame we take the projections of both members of the vector equation along the three axes of the frame. The projections are different in the different frames but the vector equation is the same, it is therefore written in covariant form.
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Lacava, F. (2016). Relativistic Covariance of Electrodynamics. In: Classical Electrodynamics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-39474-9_7
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DOI: https://doi.org/10.1007/978-3-319-39474-9_7
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