We’re given an electromagnetic wave in terms
of the electric field.
Our task is to find the direction of the magnetic
field.
In an electromagnetic wave, the electric field,
the magnetic field and the wave vector k are
all perpendicular to each other.
The electric field points in the direction
(1,1,0) and the wave vector in the direction
(0,0,1).
This is obvious from the terms within the
sine, since for a wave there’s the inner
product of the wave vector with the position
vector.
And the only possibility that the result yields
k times z is that k_x and k_y are zero.
We now have to find a direction that is orthogonal
to both (1,1,0) and (0,0,1).
If we call this direction of the magnetic
field (a,b,c), then we immediately see that
c has to equal zero.
And that a has to be negative b.
So we have to look for an answer that has
no e_z and where e_x and e_y have opposite
sign.
This only leaves answer (B) as the correct
one.
The important concept in this problem is that
in an electromagnetic wave, E, B and k are
orthogonal to each other.
That’s pretty much it, thanks for watching!
