- #1
andresordonez
- 68
- 0
When dealing with electrodynamics it is usual to use complex variables for the electromagnetic field while taking into account that the electromagnetic field is real and that at the end one has to take the real part of the complex solution for the field. However, what happens to compound expressions like the energy
[tex]
E = \frac{1}{2}(\epsilon \vec{E}^2 + \mu \vec{H}^2)
[/tex]
or the Poynting vector
[tex]
\vec{S} = \vec{E}\times\vec{H}
[/tex]
?
For the energy I see two options, either I stick with the real part of the field
[tex]
E = \frac{1}{2}(\epsilon \left[\Re(\vec{E})\right]^2 + \mu \left[\Re(\vec{H})\right]^2)
[/tex]
or I use a more general definition of the energy
[tex]
E = \frac{1}{2}(\epsilon \left|\vec{E}\right|^2 + \mu \left|\vec{H}\right|^2)
[/tex]
Note these two definitions for the energy are different since if
[tex]
z=x+iy
[/tex]
then
[tex]
\Re(z)^2 = x^2
[/tex]
[tex]
\left|z\right|^2 = x^2+y^2
[/tex]
For the Poynting vector there will be even more options since it is not as symmetric as the energy
[tex]
\vec{S} = \Re(\vec{E})\times\Re(\vec{H})
[/tex]
[tex]
\vec{S} = \vec{E} \times \vec{H}^*
[/tex]
[tex]
\vec{S} = \vec{E}^* \times \vec{H}
[/tex]
[tex]
\vec{S} = \left|\vec{E}\right| \times \left|\vec{H}\right|
[/tex]
My initial guess was that one should use only the real part of the field when calculating these compound energies (i.e. use the first options), but then I was reading Jackson's electrodynamics and I found an expression for the Poynting vector like in the second option, i.e.
[tex]
\vec{S} = \vec{E} \times \vec{H}^*
[/tex]
and an expression for the energy like in the second option as well, i.e.
[tex]
E = \frac{1}{2} \times (\epsilon \left|\vec{E}\right|^2 + \mu \left|\vec{H}\right|^2)
[/tex]
I suppose that the expressions given by Jackson are the right ones but I really don't understand why. For example why isn't the Poynting vector calculated as in the third option instead? or why should I take into account the imaginary part of the fields if they are just a convenience for the calculations? Are the imaginary parts more than just an instrument for the calculations? Is Jackson wrong?
I'd really be very grateful to anyone who could clarify this to me.
[tex]
E = \frac{1}{2}(\epsilon \vec{E}^2 + \mu \vec{H}^2)
[/tex]
or the Poynting vector
[tex]
\vec{S} = \vec{E}\times\vec{H}
[/tex]
?
For the energy I see two options, either I stick with the real part of the field
[tex]
E = \frac{1}{2}(\epsilon \left[\Re(\vec{E})\right]^2 + \mu \left[\Re(\vec{H})\right]^2)
[/tex]
or I use a more general definition of the energy
[tex]
E = \frac{1}{2}(\epsilon \left|\vec{E}\right|^2 + \mu \left|\vec{H}\right|^2)
[/tex]
Note these two definitions for the energy are different since if
[tex]
z=x+iy
[/tex]
then
[tex]
\Re(z)^2 = x^2
[/tex]
[tex]
\left|z\right|^2 = x^2+y^2
[/tex]
For the Poynting vector there will be even more options since it is not as symmetric as the energy
[tex]
\vec{S} = \Re(\vec{E})\times\Re(\vec{H})
[/tex]
[tex]
\vec{S} = \vec{E} \times \vec{H}^*
[/tex]
[tex]
\vec{S} = \vec{E}^* \times \vec{H}
[/tex]
[tex]
\vec{S} = \left|\vec{E}\right| \times \left|\vec{H}\right|
[/tex]
My initial guess was that one should use only the real part of the field when calculating these compound energies (i.e. use the first options), but then I was reading Jackson's electrodynamics and I found an expression for the Poynting vector like in the second option, i.e.
[tex]
\vec{S} = \vec{E} \times \vec{H}^*
[/tex]
and an expression for the energy like in the second option as well, i.e.
[tex]
E = \frac{1}{2} \times (\epsilon \left|\vec{E}\right|^2 + \mu \left|\vec{H}\right|^2)
[/tex]
I suppose that the expressions given by Jackson are the right ones but I really don't understand why. For example why isn't the Poynting vector calculated as in the third option instead? or why should I take into account the imaginary part of the fields if they are just a convenience for the calculations? Are the imaginary parts more than just an instrument for the calculations? Is Jackson wrong?
I'd really be very grateful to anyone who could clarify this to me.