The LaTeX: yy-component of the electric field of an electromagnetic wave traveling in the LaTeX: +x+ x-direction through vacuum obeys the equation LaTeX: E_y= (375\;{\rm N/C}) \cos[kx - (2.20\!\times\!10^{14}\;{\rm rad/s})t]E y = ( 375 N / C ) cos ⁡ [ k x − ( 2.20 × 10 14 r a d / s ) t ]. What is the wavelength of this electromagnetic wave in micrometers?

Explanation:

Given that,

Wave equation is

Ey = 375•Cos(kx — 2.2 × 10¹⁴t)

Generally,

The wave equation is written as

Ey = Eo•Cos(kx-wt)

Where

K is wave number

ω is the angular frequency

So, comparing this to the the given wave equation, we found out that,

ω = 2.2 × 10¹⁴ rad / s

Then,

ω = 2πf

f = ω / 2π

From the relationship between wavelength and speed

v = fλ

Substitute for f

v = λ × ω / 2π

In electromagnetic we are dealing with speed of light , then, v = c

c = λ × ω / 2π

c is speed of light in vacuum and it's value is

c = 3 × 10^8 m/s

Make wavelength subject of the formula

λ = 2π × c / ω

λ = 2π × 3 × 10^8 / 2.2 × 10¹⁴

λ = 8.57 × 10^-6 m

λ = 8.57 μm

The wave length is 8.57 μm