Question

Write Maxwell equation for harmonically varying field.

Answer 1

Harmonically varying field can be written using the Euler's law:

$E=E_0 e^{i \omega t}$

$H=H_0 e^{i \omega t}$

Maxwell's third and forth equations in vacuum:

$\displaystyle \nabla \times E = -\frac{1}{c}\frac{\partial B}{\partial t}$

$\displaystyle \nabla \times B = \frac{1}{c} \frac{\partial E}{\partial t}$

$\displaystyle \frac{\partial B}{\partial t}= i\omega B$

$\displaystyle \frac{\partial E}{\partial t}= i\omega E$

Then Maxwell equations for harmonically varying field looks following

$\displaystyle \nabla \times E = -\frac{1}{c} i \omega B$

$\displaystyle \nabla \times B = \frac{1}{c} i \omega E$