A plane electromagnetic wave propagating in x-direction is described by
E_y=(200 Vm⁻¹) sin [1.5 × 10⁷ t-0.05 x] ;
The intensity of the wave is :
(Use ε₀=8.85 × 10⁻¹² C² N⁻¹ m⁻²)

Solution

Physics sir · 4 steps

Pick the amplitude out of the equation, then use I = 1/2ε₀ c E₀². Do not drop the half.

1. Read the wave. Compare with E_y = E₀ sin(ω t - kx). The amplitude is E₀ = 200 V m⁻¹, with ω = 1.5 × 10⁷ rad s⁻¹ and k = 0.05 m⁻¹.
2. Wave speed. v = ω/k = (1.5 × 10⁷)/0.05 = 3 × 10⁸ m s⁻¹. That is c, so the wave travels in vacuum.

   I = 1/2ε₀ c E₀²

3. Put in numbers. I = 1/2 × 8.85 × 10⁻¹² × 3 × 10⁸ × (200)² = 1/2 × 106.2 = 53.1 W m⁻².
4. So. The intensity is 53.1 W m⁻², which is option (b).

The idea to keep. Intensity is the time-averaged energy density times c. The average of sin² over a cycle is 1/2, so the formula is 1/2ε₀ c E₀² with the peak field E₀.

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