Magnetic field=?

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1 Answer
Mar 4, 2018

#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}#