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
\nabla\times E = -dB/dt
I want to be able to get
\nabla\times \vec{E} = -j\omega\vec{B}
where
E = time-varying form of E-field
B = time-varying form of applied magnetic field (is this the correct assumption ?)
\vec{E} = time-harmonic form of E field
\vec{B} = 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
E = Re(\vec{E}ej*\omega*t) (1.1)
B = Re(\vec{B}ej*\omega*t) (1.2)
where Re() = real part of complex number
However, if this is the case, E = \vec{E}cos(\omega*t) will be true and given that there is a time-derivative to the RHS of the time-varying equation (which is going to produce dB/dt = -\omega*\vec{B}sin(\omega*t)), 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
B = \vec{B}ej*\omega*t
term and
eliminates the common e(j*\omega*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
\nabla\times E = -dB/dt
I want to be able to get
\nabla\times \vec{E} = -j\omega\vec{B}
where
E = time-varying form of E-field
B = time-varying form of applied magnetic field (is this the correct assumption ?)
\vec{E} = time-harmonic form of E field
\vec{B} = 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
E = Re(\vec{E}ej*\omega*t) (1.1)
B = Re(\vec{B}ej*\omega*t) (1.2)
where Re() = real part of complex number
However, if this is the case, E = \vec{E}cos(\omega*t) will be true and given that there is a time-derivative to the RHS of the time-varying equation (which is going to produce dB/dt = -\omega*\vec{B}sin(\omega*t)), 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
B = \vec{B}ej*\omega*t
term and
eliminates the common e(j*\omega*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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