Electromagnetism - Superposition

Questions 54-55 concern a plane electromagnetic wave that is a superposition of two independent orthogonal plane waves and can be written as the real part of 

 E = x̂ E₁ exp [i(kz −ωt)] +  ŷ E₂ exp [i(kz − ωt + π)]
where k, ω, E₁ and E₂ are real

If E₂ = E₁, the tip of the electric field vector will describe a trajectory that, as viewed along the z-axis from positive z and looking toward the origin, is a

A. Line at 45^o to the + x-axis
B. Line at 135^o to the + x-axis
C. Clockwise circle
D. Counterclockwise circle
E. Random path

(GR9677 #54)

Solution:

E = x̂ E₁ e^{i(kz − ωt)}  +  ŷ E₂ e^{i(kz − ωt +π)} 
E = x̂ E₁ e^{i(kz − ωt)}  +  ŷ E₂ e^{i(kz − ωt)} · e^iπ 

with 
E₂ = E₁ = E
e^iπ = −1

E = E e^{i(kz − ωt)} x̂ − E e^{i(kz − ωt)} ŷ 
E = a x̂ − a ŷ 

tan θ  = a / (−a) = −1

tan 45  = 1
tan 135 = tan 315 = −1

Answer: B


Note: 
e^iϕ = cos ϕ + isin ϕ
e^iπ = cos π + isin π =  −1 + 0 = −1