An electromagnetic wave travels in vacuum along z - direction : `vec(E )=(E_(1)hat(i)+E_(2)hat(j))cos(kz-omega t)`. Choose the correct options from the following :

Options

A. The associated magnetic field is given as `vec(B)=(1)/(C )hat(k)xx vec(E )=(1)/(omega)(hat(k)xx vec(E ))`
B. The electromagnetic field can be written in terms of the associated magnetic field as `vec(E )=C(vec(B)xx hat(k))`
C. `hat(k).vec(E )=0, hat(k).vec(B)=0`
D. `hat(k)xx vec(E )=0, hat(k)xx vec(B)=0`

Answer

A, B, C

Explanation

Let electromagnetic wave propagate in negative z - direction. (Hence, its electric field is given by `vec(E )=E_(0)cos(kz - omega t)` which is perpendicular to z - direction and in negative y - direction.
In electromagnetic waves corresponding magnetic field `vec(B)` is in x - direction `hat(k)xx vec(E )`.
Because `B_(0)=(E_(0))/(C )`
`therefore B=(1)/(C )(hat(k)xx vec(E ))`
Electric field can be written in form of corresponding magnetic field as `vec(E )=C(vec(B)xx hat(k))`.
Angle between `hat(k)` and `vec(E )` is `90^(@)` and angle between `hat(k)` and `ve(B)` is also `90^(@)`.
So, `vec(E )=(1)E cos 90^(@)=0[because hat(k)=1]` and `hat(k).vec(B)=1B cos 90^(@)`
`therefore hat(k).vec(B)=0`