g An electromagnetic wave is traveling on a transmission line. Its voltage is given by v(x, t) = 10 Exp[−γx] cos(15x − 109 t) where x is the distance (in meters) from the generator, and t is in seconds. (a) What is the frequency, wavelength and phase veolicty of the electromagnetic wave. (b) at x = 3 m, the wave amplitude is measured to be 4V . Determine γ

a)

f  = 1.59*10⁸ Hz.

λ = 0.42 m

vp = 0.67*10⁸ m/s = 0.22 c

b)

γ = 0.31 1/m

Explanation:

a)

In a transmission line, voltage varies sinusoidally with the distance and time.We can express mathematically this relationship in a cosine form as follows:V (x,t) = A cos (kx-ωt + φ₀) (1)where A = amplitude of the wave (Volts)k = number of wave = 2*π/λ (being λ the wavelength of the wave)ω = angular frequency = 2*π*f (being f the frequency of the wave)φ₀ = phase of the wave.In our case, the expression for the voltage is as follows:v (x,t) = 10 V Exp [-γx] cos (15x-10⁹ t) (2)From (1) and (2) we find the following equalities:k = 15 = 2*π/λ (3)Solving for λ:

       

ω = 2*π*f = 10⁹ rad/sec (5)Solving for f:

       

The phase velocity of the wave is just the product of the wavelength times the frequency, as follows:

       

b)

We can express the amplitude at any time as follows:

       

If we know that V= 4 V at x=3m, (8) becomes:

       

Taking ln on both sides, we can solve for γ as follows:

         

       γ = 0.31 1/m

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