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Question:

A plane polarized monochromatic EM wave is travelling in a vacuum along the z-direction such that at t=t1, it is found that the electric field is zero at a spatial point z1. The next zero that occurs in its neighborhood is at z2. The frequency of the electromagnetic wave is:

3×10⁸|z₂−z₁|

6×10⁸|z₂−z₁|

1.5×10⁸|z₂−z₁|

1t₁+|z₂−z₁|3×10⁸

Solution:

E = E₀eⁱ(kz-ωt)
At t = t₁, z = z₁, E = 0, the next zero that occurs in its neighborhood is at z₂.
The frequency of the electromagnetic wave at t₂
eⁱ(kz₁-ωt₁) = eⁱ(kz₂-ωt₂)
kz₁ - ωt₁ = kz₂ - ωt₂
(t₂ - t₁)ω = k(z₂ - z₁)
where k = 2π/λ, ω = 2πν
(t₂ - t₁) = 2πν(z₂ - z₁)/2π/λ = λν/c (z₂ - z₁)
(t₂ - t₁) = λ/c(z₂ - z₁)
λ = c(t₂ - t₁)/(z₂ - z₁)
Frequency is f ≈ 1/t then 1/(t₂ - t₁) = c/(z₂ - z₁)
Frequency = 3×10⁸/(|z₂ - z₁|)