Question

Magnetic field=?

Answer 1

`vec{B} = 2{E_0 }/c sin kz sin omega t hat{k}`

Explanation:

We will use Faraday's law :

`nabla times vec{E} + {partial \vec{B}}/{partial t} = 0`

to determine `\vec{B}`. We first calculate the curl of `\vec{E}` :

`nabla times vec{A} = | (hat i, hat j, hat k),(partial/{partial x}, partial/{partial y}, partial/{partial z}),(2E_0 cos kz cos omega t,0,0)|`
`= hat j partial/{partial z }(2E_0 cos kz cos omega t) = -2E_0 k sin kz cos omega t hat{k}`

Thus

`{partial vec{B}}/{partial t} = 2E_0 k sin kz cos omega t hat{k}`

Integration yields

`vec B = 2{E_0 k}/omega sin kz sin omega t hat{k} = 2{E_0 }/c sin kz sin omega t hat{k}`