In a plane electromagnetic wave, the directions of electric field and magnetic field are represented by k̂ and 2î - 2ĵ, respectively. What is the unit vector along direction of propagation of the wave?

Solution

Physics sir · 4 steps

The wave always travels along E⃗ × B⃗, so take that cross product and normalise it.

1. Direction rule. In a plane EM wave, E⃗, B⃗ and the direction of travel are mutually perpendicular. The wave moves along E⃗ × B⃗, in that order.

   k̂ × (2î - 2ĵ) = 2(k̂ × î) - 2(k̂ × ĵ) = 2ĵ + 2î

2. Use the cyclic order. Here k̂ × î = ĵ and k̂ × ĵ = -î, so the two minus signs combine to give +2î.
3. Normalise. The magnitude of 2î + 2ĵ is √(4 + 4) = 2√2. Dividing gives 1/√2(î + ĵ).
4. So. The wave propagates along 1/√2(î + ĵ), which is option (c). As a check, it has zero dot product with both k̂ and 2î - 2ĵ.

The idea to keep. Propagation direction is E⃗ × B⃗, never B⃗ × E⃗. A quick dot-product check rules out any option that is not perpendicular to both fields.

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