Respuesta :
Answer:
[tex]B=-\dfrac{E_o}{c}cos(kx)sin(\omega t)\ k[/tex]
Explanation:
The electric field of a plane standing electromagnetic wave in a vacuum is given by :
[tex]E_y=E_o\ sin(kx)cos(\omega t)[/tex]
We need to find the corresponding expression for the magnetic field. According to equation of Maxwell's :
[tex]\bigtriangledown \times E=-\dfrac{\partial B}{\partial t}[/tex]
[tex]\bigtriangledown \times E=\begin{vmatrix}i & j & k\\ \dfrac{\partial}{\partial x} & \dfrac{\partial }{\partial y} & \dfrac{\partial}{\partial z}\\ 0& E_o\ sin(kx)cos(\omega t) & 0 \end{vmatrix}[/tex]
[tex]\bigtriangledown \times E=k[E_ok\ cos(kx)cos(\omega t)]=-\dfrac{\partial B}{\partial t}[/tex]
[tex]B=\int\limits {kE_ok\ cos(kx)cos(\omega t).dt}[/tex]
[tex]B=-\dfrac{E_ok}{\omega}cos(kx)sin(\omega t)[/tex]
Since, [tex]\omega=ck[/tex]
[tex]B=-\dfrac{E_o}{c}cos(kx)sin(\omega t)\ k[/tex]
So, the corresponding expression for the magnetic field is [tex]-\dfrac{E_o}{c}cos(kx)sin(\omega t)\ k[/tex]. Hence, this is the required solution.
