Question

A plane electromagentic wave `E=E_(m) cos ( omegat - kr) ` propagates in vacuum. Assuming the vectors `E_(m)` and `k` to be known, find the vector `H` as a function of time `t` at the point with radius vector `r=0` .

Answer 1

`vec(nabla)xxvec(E)=-(deltavec(B))/( deltat)=- mu _(0)(deltavec(H))/( deltat)`
`=nablacos ( omegat - vec(k). vec(r))xxvec(E)_(m) =vec(k) xx vec(E)_(m) sin ( omegat- vec(k). vec(r))`
At ` vec (r)=0`
`(deltavec(H))/(deltat)=-(vec(k) xxvec(E)_(m))/( mu_(0))sin omegat`
So integrating `(` ignoring a constant `)` and using `c=(1)/(sqrt(epsilon_(0)mu_(0)))`
` vec (H)= (vec(k) xx vec(E)_(m))/( mu_(0))cos c k t xx(1)/(ck)=sqrt((epsilon_(0))/( mu_(0)))(vec(k)xxvec(E)_(m))/( k ) cos c k t `