- #1
JamesGoh
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Hi all
Im trying to derive the time-harmonic form of the Maxwell equations from the original, time-dependent form, however I am not sure if my working and logic is correct
e.g. for
[tex]\nabla\times E = -dB/dt[/tex]
I want to be able to get
[tex]\nabla\times \vec{E}[/tex] = [tex]-j\omega\vec{B}[/tex]
where
[tex]E[/tex] = time-varying form of E-field
[tex]B[/tex] = time-varying form of applied magnetic field (is this the correct assumption ?)
[tex]\vec{E}[/tex] = time-harmonic form of E field
[tex]\vec{B}[/tex] = time-harmonic form of applied magnetic field
The textbook has given the relationship between the time-varying and time-harmonic forms of the field as follows
[tex]E = [/tex] Re([tex]\vec{E}[/tex]ej*[tex]\omega[/tex]*t) (1.1)
[tex]B = [/tex] Re([tex]\vec{B}[/tex]ej*[tex]\omega[/tex]*t) (1.2)
where Re() = real part of complex number
However, if this is the case, [tex]E = \vec{E}cos(\omega*t)[/tex] will be true and given that there is a time-derivative to the RHS of the time-varying equation (which is going to produce [tex]dB/dt = -\omega*\vec{B}sin(\omega*t)[/tex]), it would be impossible to get the time-harmonic form, as you have cosine on one side and sine() on the other.
The textbook simply substitutes the time-harmonic version over its time-varying version and differentiates the
[tex]B = [/tex] [tex]\vec{B}[/tex]ej*[tex]\omega[/tex]*t
term and
eliminates the common e(j*[tex]\omega[/tex]*t) on both sides, however I cannot seem to see the link between this and the equations linking the time-varying and time-harmonic forms, What could I possibly be doing wrong or misunderstand ?
Im trying to derive the time-harmonic form of the Maxwell equations from the original, time-dependent form, however I am not sure if my working and logic is correct
e.g. for
[tex]\nabla\times E = -dB/dt[/tex]
I want to be able to get
[tex]\nabla\times \vec{E}[/tex] = [tex]-j\omega\vec{B}[/tex]
where
[tex]E[/tex] = time-varying form of E-field
[tex]B[/tex] = time-varying form of applied magnetic field (is this the correct assumption ?)
[tex]\vec{E}[/tex] = time-harmonic form of E field
[tex]\vec{B}[/tex] = time-harmonic form of applied magnetic field
The textbook has given the relationship between the time-varying and time-harmonic forms of the field as follows
[tex]E = [/tex] Re([tex]\vec{E}[/tex]ej*[tex]\omega[/tex]*t) (1.1)
[tex]B = [/tex] Re([tex]\vec{B}[/tex]ej*[tex]\omega[/tex]*t) (1.2)
where Re() = real part of complex number
However, if this is the case, [tex]E = \vec{E}cos(\omega*t)[/tex] will be true and given that there is a time-derivative to the RHS of the time-varying equation (which is going to produce [tex]dB/dt = -\omega*\vec{B}sin(\omega*t)[/tex]), it would be impossible to get the time-harmonic form, as you have cosine on one side and sine() on the other.
The textbook simply substitutes the time-harmonic version over its time-varying version and differentiates the
[tex]B = [/tex] [tex]\vec{B}[/tex]ej*[tex]\omega[/tex]*t
term and
eliminates the common e(j*[tex]\omega[/tex]*t) on both sides, however I cannot seem to see the link between this and the equations linking the time-varying and time-harmonic forms, What could I possibly be doing wrong or misunderstand ?
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