Hello :smile:!
Duality in electromagnetism is possible in the following way. Let [itex]\mathbf{E}, \mathbf{H}[/itex] be a field which by hypohtesis is a solution of the source-free Maxwell's equations (for phasors)
(1)
[itex]\nabla \times \mathbf{E} = - j \omega \mu \mathbf{H}[/itex]
[itex]\nabla \times \mathbf{H} = j \omega \epsilon \mathbf{E}[/itex]
If we systematically substitute
[itex]\mathbf{E}[/itex] with [itex]\mathbf{H}[/itex]
[itex]\mathbf{H}[/itex] with [itex]-\mathbf{E}[/itex]
[itex]\epsilon[/itex] with [itex]\mu[/itex]
[itex]\mu[/itex] with [itex]\epsilon[/itex]
we obtain another set of equations, but we accept them because we already know them and we already know that they are verified (they are the same as above):
(2)
[itex]\begin{equation} \nabla \times \mathbf{H} = j \omega \epsilon \mathbf{E} \end{equation}[/itex]
[itex]\nabla \times \mathbf{E} = - j \omega \mu \mathbf{H}[/itex]
If the SI units are used, all the substitutions are needed.
But with sources this is no more true: that is, the original Maxwell's equation would be
[itex]\nabla \times \mathbf{E} = - j \omega \mu \mathbf{H}[/itex]
[itex]\nabla \times \mathbf{H} = \mathbf{J} + j \omega \epsilon \mathbf{E}[/itex]
and the new equations with the 4 substitutions become
(3)
[itex]\nabla \times \mathbf{H} = j \omega \epsilon \mathbf{E}[/itex]
[itex]\nabla \times \mathbf{E} = - \mathbf{J} - j \omega \mu \mathbf{H}[/itex]
Duality is intended also in that form. But now the last equation does no more coincide with the first equation of the set (1). How can we accept it and say (like textbooks) that a new magnetic current [itex]\mathbf{J}_m = \mathbf{J}[/itex] must be introduced? And how can we be sure that this "new" magnetic current produces the field [itex]\mathbf{H}, \mathbf{-E}[/itex] if we can't verify the equations (3)?
Thank you for having read,
Emily
Duality in electromagnetism is possible in the following way. Let [itex]\mathbf{E}, \mathbf{H}[/itex] be a field which by hypohtesis is a solution of the source-free Maxwell's equations (for phasors)
(1)
[itex]\nabla \times \mathbf{E} = - j \omega \mu \mathbf{H}[/itex]
[itex]\nabla \times \mathbf{H} = j \omega \epsilon \mathbf{E}[/itex]
If we systematically substitute
[itex]\mathbf{E}[/itex] with [itex]\mathbf{H}[/itex]
[itex]\mathbf{H}[/itex] with [itex]-\mathbf{E}[/itex]
[itex]\epsilon[/itex] with [itex]\mu[/itex]
[itex]\mu[/itex] with [itex]\epsilon[/itex]
we obtain another set of equations, but we accept them because we already know them and we already know that they are verified (they are the same as above):
(2)
[itex]\begin{equation} \nabla \times \mathbf{H} = j \omega \epsilon \mathbf{E} \end{equation}[/itex]
[itex]\nabla \times \mathbf{E} = - j \omega \mu \mathbf{H}[/itex]
If the SI units are used, all the substitutions are needed.
But with sources this is no more true: that is, the original Maxwell's equation would be
[itex]\nabla \times \mathbf{E} = - j \omega \mu \mathbf{H}[/itex]
[itex]\nabla \times \mathbf{H} = \mathbf{J} + j \omega \epsilon \mathbf{E}[/itex]
and the new equations with the 4 substitutions become
(3)
[itex]\nabla \times \mathbf{H} = j \omega \epsilon \mathbf{E}[/itex]
[itex]\nabla \times \mathbf{E} = - \mathbf{J} - j \omega \mu \mathbf{H}[/itex]
Duality is intended also in that form. But now the last equation does no more coincide with the first equation of the set (1). How can we accept it and say (like textbooks) that a new magnetic current [itex]\mathbf{J}_m = \mathbf{J}[/itex] must be introduced? And how can we be sure that this "new" magnetic current produces the field [itex]\mathbf{H}, \mathbf{-E}[/itex] if we can't verify the equations (3)?
Thank you for having read,
Emily