The electric field of a plane electromagnetic wave is given by
E⃗ = E₀( x + y )sin ( kz - ω t )
Its magnetic field will be given by :

Solution

Physics sir · 4 steps

The wave runs along +z, so B⃗ must be z × E⃗ divided by c. Do that one cross product.

1. Direction of travel. The phase is kz - ω t. A constant phase moves to larger z as t grows, so the wave travels along +z.
2. Rule for B. In a plane EM wave, B⃗ is in phase with E⃗, has magnitude E/c, and is perpendicular to both E⃗ and the direction of travel. All of this is packed into one relation.

   B⃗ = 1/c z × E⃗ = E₀/c(z × x + z × y)sin(kz - ω t)

3. Cross products. z × x = y and z × y = -x. So B⃗ = E₀/c(-x + y)sin(kz - ω t).
4. Check. (x + y) × (-x + y) = z + z = 2z. So E⃗ × B⃗ points along +z, the direction of travel. This confirms option (d).

The idea to keep. For a plane EM wave, B⃗ = 1/cn × E⃗, where n is the direction of travel. Then check that E⃗ × B⃗ points along n.

Answer from the NTA final answer key. Solution by PrepWiser.