Final answer:
The oscillating electric field equation is completed by including a phase argument that is the dot product of wave vector k and the position vector, resulting in vec{E} = E₀ sin(ω t - k[(x+y)/√2])[(^-i + ^j)/√2].
Step-by-step explanation:
To write the equation of an oscillating electric field (EF) that is propagating along a specific direction, you must substitute the correct phase argument into the sine function. In your case, the oscillating electric field is propagating along the direction [(^i+^j)/√2], which is a vector with equal components in the x and y directions, normalized to unit length.
Considering the electric field vector vec{E} = E₀ sin(ω t - k( ))[(^-i + ^j)/√2], the appropriate phase argument that goes into the empty bracket is a dot product of the wave vector k with the position vector (x^i + y^j), since the wave is propagating along both x and y axes.
Therefore, the full equation of the electric field with the phase argument filled in becomes: vec{E} = E₀ sin(ω t - k[(x+y)/√2])[(^-i + ^j)/√2]. This represents a plane wave propagating diagonally in the x-y plane with an electric field oscillating in the direction perpendicular to the propagation direction, normalized to unit length.